Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Precalculus
In Problems 17-28, write down the first five terms of each sequence. { s n }={ ( 4 3 ) n }In Problems 17-28, write down the first five terms of each sequence. { t n }={ ( 1 ) n ( n+1 )( n+2 ) }In Problems 17-28, write down the first five terms of each sequence. { a n }={ 3 n n }In Problems 17-28, write down the first five terms of each sequence. { b n }={ n e n }In Problems 17-28, write down the first five terms of each sequence. { c n }={ n 2 2 n }In Problems 29-36, the given pattern continues. Write down the nth term of a sequence { a n } suggested by the pattern. 1 2 , 2 3 , 3 4 , 4 5 ,...28AYUIn Problems 29-36, the given pattern continues. Write down the nth term of a sequence { a n } suggested by the pattern. 1, 1 2 , 1 4 , 1 8 ,...In Problems 29-36, the given pattern continues. Write down the nth term of a sequence { a n } suggested by the pattern. 2 3 , 4 9 , 8 27 , 16 81 ,...In Problems 29-36, the given pattern continues. Write down the nth term of a sequence { a n } suggested by the pattern. 1,1,1,1,1,1,...In Problems 29-36, the given pattern continues. Write down the nth term of a sequence { a n } suggested by the pattern. 1, 1 2 ,3, 1 4 ,5, 1 6 ,7, 1 8 ,...In Problems 29-36, the given pattern continues. Write down the nth term of a sequence { a n } suggested by the pattern. 1,2,3,4,5,6,...In Problems 29-36, the given pattern continues. Write down the nth term of a sequence { a n } suggested by the pattern. 2,4,6,8,10,...In Problems 37-50, a sequence is defined recursively. Write down the first five terms. a 1 =2 ; a n =3+ a n1In Problems 37-50, a sequence is defined recursively. Write down the first five terms. a 1 =3 ; a n =4 a n1In Problems 37-50, a sequence is defined recursively. Write down the first five terms. a 1 =2 ; a n =n+ a n1In Problems 37-50, a sequence is defined recursively. Write down the first five terms. a 1 =1 ; a n =n a n1In Problems , a sequence is defined recursively. List the first five terms.
In Problems 37-50, a sequence is defined recursively. Write down the first five terms. a 1 =2 ; a n = a n1In Problems 37-50, a sequence is defined recursively. Write down the first five terms. a 1 =3 ; a n = a n1 nIn Problems 37-50, a sequence is defined recursively. Write down the first five terms. a 1 =2 ; a n =n+3 a n1In Problems 37-50, a sequence is defined recursively. Write down the first five terms. a 1 =1 ; a 2 =2 ; a n = a n1 a n2In Problems 37-50, a sequence is defined recursively. Write down the first five terms. a 1 =1 ; a 2 =1 ; a n = a n2 +n a n1In Problems 37-50, a sequence is defined recursively. Write down the first five terms. a 1 =A ; a n = a n1 +dIn Problems 37-50, a sequence is defined recursively. Write down the first five terms. a 1 =A ; a n =r a n1 ; r0In Problems 37-50, a sequence is defined recursively. Write down the first five terms. a 1 = 2 ; a n = 2+ a n1In Problems 37-50, a sequence is defined recursively. Write down the first five terms. a 1 = 2 ; a n = a n1 2In Problems 51-60, write out each sum. k=1 n ( k+2 )In Problems 51-60, write out each sum. k=1 n ( 2k+1 )In Problems 51-60, write out each sum. k=1 n k 2 2In Problems 51-60, write out each sum. k=1 n ( k+1 ) 2In Problems 51-60, write out each sum. k=0 n 1 3 kIn Problems 51-60, write out each sum. k=0 n ( 3 2 ) kIn Problems 51-60, write out each sum. k=0 n1 1 3 k+1In Problems 51-60, write out each sum. k=0 n1 ( 2k+1 )In Problems 51-60, write out each sum. k=2 n ( 1 ) k lnkIn Problems 51-60, write out each sum. k=3 n ( 1 ) k+1 2 kIn Problems 61-70, express each sum using summation notation. 1+2+3+...+20In Problems 61-70, express each sum using summation notation. 1 3 + 2 3 + 3 3 +...+ 8 3In Problems 61-70, express each sum using summation notation. 1 2 + 2 3 + 3 4 +...+ 13 13+1In Problems 61-70, express each sum using summation notation. 1+3+5+7+...+[ 2( 12 )1 ]In Problems 61-70, express each sum using summation notation. 1 1 3 + 1 9 1 27 +...+ ( 1 ) 6 ( 1 3 6 )In Problems 61-70, express each sum using summation notation. 2 3 4 9 + 8 27 ...+ ( 1 ) 12 ( 2 3 ) 11In Problems 61-70, express each sum using summation notation. 3+ 3 2 2 + 3 3 3 +...+ 3 n nIn Problems 61-70, express each sum using summation notation. 1 e + 2 e 2 + 3 e 3 +...+ n e nIn Problems 61-70, express each sum using summation notation. a+( a+d )+( a+2d )+...+( a+nd )In Problems 61-70, express each sum using summation notation. a+ar+a r 2 +...+a r n1In Problems 71-82, find the sum of each sequence. k=1 40 5In Problems 71-82, find the sum of each sequence. k=1 50 8In Problems 71-82, find the sum of each sequence. k=1 40 kIn Problems 71-82, find the sum of each sequence. k=1 24 ( k )In Problems 71-82, find the sum of each sequence. k=1 20 ( 5k+3 )In Problems 71-82, find the sum of each sequence. k=1 26 ( 3k7 )In Problems 71-82, find the sum of each sequence. k=1 16 ( k 2 +4 )In Problems 71-82, find the sum of each sequence. k=0 14 ( k 2 4 )In Problems 71-82, find the sum of each sequence. k=10 60 ( 2k )In Problems 71-82, find the sum of each sequence. k=8 40 ( 3k )In Problems 71-82, find the sum of each sequence. k=5 20 k 3In Problems 71-82, find the sum of each sequence. k=4 24 k 3Credit Card Debt John has a balance of on his Discover card, which charges interest per month on any unpaid balance from the previous month. John can afford to pay $100 toward the balance each month. His balance each month after making a payment is given by the recursively defined sequence
.
Determine John's balance after making the first payment. That is, determine .
Trout Population A pond currently contains 2000 trout. A fish hatchery decides to add 20 trout each month. It is also known that the trout population is growing at a rate of 3 per month. The size of the population after n months is given by the recursively defined sequence p0=3000pn=1.03pn1+20. How many trout are in the pond after 2 months? That is. what is p2?Car Loans Phil bought a car by taking out a loan for 18,500 at 0.5 interest per month. Phils normal monthly payment is 434.47 per month, but he decides that he can afford to pay 100 extra toward the balance each month. His balance each month is given by the recursively defined sequence B0=18,500Bn=1.005Bn1534.47. Determine Phils balance alter making the first payment. That is. determine B1.Environmental Control The Environmental Protection Agency (EPA) determines that Maple Lake has tons of pollutant as a result of industrial waste and that of the pollutant present is neutralized by solar oxidation every year. The EPA imposes new pollution control laws that result in tons of new pollutant entering the lake each year. The amount of pollutant in the lake after years is given by the recursively defined sequence
.
Determine the amount of pollutant in the lake after years. That is, determine .
Growth of a Rabbit Colony A colony of rabbits begins with one pair of mature rabbits, which produces a pair of offspring (one male, one female) each month. Assume that all rabbits mature in month and produce a pair of offspring (one male, one female) after months. If no rabbits ever die, how many pairs of mature rabbits are there aftermonths? See figure. top right. (Hint: A Fibonacci sequence models this colony. Do you see why?)
86AYUThe Pascal Triangle The triangular array shown, called the Pascal triangle, is partitioned using diagonal lines as shown. Find the sum of the numbers in each diagonal row. Do you recognize this sequence?88AYU89AYU90AYU91AYU92AYU93AYU94AYU95AYU96AYU97AYU98AYU99AYU100AYU101AYUIn a(n) _________ sequence, the difference between successive terms is a constant.2AYUIf the 5th term of an arithmetic sequence is 12 and the common difference is 5, then the 6th term of the sequence is _______.True or False The sum S n of the first n terms of an arithmetic sequence { a n } whose first term is a 1 can be found using the formula S n = n 2 ( a 1 + a n ) .In Problems 7-16, show that each sequence is arithmetic. Find the common difference and write out the first four terms. { S n }={ n+4 }In Problems 7-16, show that each sequence is arithmetic. Find the common difference and write out the first four terms. { S n }={ n5 }In Problems 7-16, show that each sequence is arithmetic. Find the common difference and write out the first four terms. { a n }={ 2n5 }In Problems 7-16, show that each sequence is arithmetic. Find the common difference and write out the first four terms. { b n }={ 3n+1 }In Problems 7-16, show that each sequence is arithmetic. Find the common difference and write out the first four terms. { c n }={ 62n }In Problems 7-16, show that each sequence is arithmetic. Find the common difference and write out the first four terms. { a n }={ 42n }In Problems 7-16, show that each sequence is arithmetic. Find the common difference and write out the first four terms. { t n }={ 1 2 1 3 n }In Problems 7-16, show that each sequence is arithmetic. Find the common difference and write out the first four terms. { t n }={ 2 3 + n 4 }In Problems 7-16, show that each sequence is arithmetic. Find the common difference and write out the first four terms. { S n }={ ln 3 n }In Problems 7-16, show that each sequence is arithmetic. Find the common difference and write out the first four terms. { S n }={ e lnn }In Problems 17-24, find the nth term of the arithmetic sequence { a n } whose initial term a 1 and common difference d are given. What is the 51st term? a 1 =2;d=3In Problems 17-24, find the nth term of the arithmetic sequence { a n } whose initial term a 1 and common difference d are given. What is the 51st term? a 1 =2;d=4In Problems , find the th term of the arithmetic sequence whose first term and common difference are given. What is the st term?
In Problems 17-24, find the nth term of the arithmetic sequence { a n } whose initial term a 1 and common difference d are given. What is the 51st term? a 1 =6;d=2In Problems 17-24, find the nth term of the arithmetic sequence { a n } whose initial term a 1 and common difference d are given. What is the 51st term? a 1 =0;d= 1 2In Problems 17-24, find the nth term of the arithmetic sequence { a n } whose initial term a 1 and common difference d are given. What is the 51st term? a 1 =1;d= 1 3In Problems 17-24, find the nth term of the arithmetic sequence { a n } whose initial term a 1 and common difference d are given. What is the 51st term? a 1 = 2 ;d= 2In Problems 17-24, find the nth term of the arithmetic sequence { a n } whose initial term a 1 and common difference d are given. What is the 51st term? a 1 =0;d=In Problems 25-30, find the indicated term in each arithmetic sequence. 100thtermof2,4,6,...In Problems 25-30, find the indicated term in each arithmetic sequence. 80thtermof1,1,3,...In Problems 2530, find the indicated term in each arithmetic sequence. 90 the term of 3,3,9,In Problems 25-30, find the indicated term in each arithmetic sequence. 80thtermof5,0,5,...In Problems 25-30, find the indicated term in each arithmetic sequence. 80thtermof2, 5 2 ,3, 7 2 ,...In Problems 25-30, find the indicated term in each arithmetic sequence. 70thtermof2 5 ,4 5 ,6 5 ,...In Problems 31-38, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 8thtermis8;20thtermis44In Problems 31-38, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 8thtermis8;20thtermis44In Problems 31-38, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 9thtermis5;15thtermis31In Problems 31-38, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 8thtermis4;18thtermis96In Problems 31-38, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 15thtermis0;40thtermis50In Problems 31-38, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 5thtermis2;13thtermis30In Problems 31-38, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 14thtermis1;18thtermis9In Problems 31-38, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 12thtermis4;18thtermis28In Problems 39-56, find each sum. 1+3+5++( 2n1 )In Problems 39-56, find each sum. 2+4+6++2nIn Problems 39-56, find each sum. 7+12+17++( 2+5n )In Problems 39-56, find each sum. 1+3+7++( 4n5 )In Problems 39-56, find each sum. 2+4+6++70In Problems 39-56, find each sum. 1+3+5++59In Problems 3956, find each sum. 951+...+39In Problems 39-56, find each sum. 2+5+8++41In Problems , find each sum.
In Problems 39-56, find each sum. 7+1511299In Problems 39-56, find each sum. 4+4.5+5+5.5++100In Problems 39-56, find each sum. 8+8 1 4 +8 1 2 +8 3 4 +9++5049AYU50AYUIn Problems 39-56, find each sum. n=1 100 ( 6 1 2 n )52AYU53AYUIn Problems 39-56, find each sum. The sum of the first 46 terms of the sequence 2,1,4,7,...55AYU56AYU57AYU58AYU59AYU60AYU61AYU62AYU63AYU64AYU65AYU66AYU67AYU68AYU69AYUIf is invested at per annum compounded semiannually, how much is in the account after years ?
2AYUIn a(n) _____________ sequence, the ratio of successive terms is a constant.4AYU5AYU6AYU7AYU8AYUIn problems 918, show that each sequence is geometric. Then find the common ratio and list the first four terms. sn=4nIn Problems 9-18, show that each sequence is geometric. Then find the common ratio and write out the first four terms. { s n }={ ( 5 ) n }11AYUIn Problems 9-18, show that each sequence is geometric. Then find the common ratio and write out the first four terms. { b n }={ ( 5 2 ) n }In Problems 9-18, show that each sequence is geometric. Then find the common ratio and write out the first four terms. { c n }={ 2 4 n1 }In Problems 9-18, show that each sequence is geometric. Then find the common ratio and write out the first four terms. { d n }={ 3 n 9 }In problems 918, show that each sequence is geometric. Then find the common ratio and list the first four terms. en=7n/4In Problems 9-18, show that each sequence is geometric. Then find the common ratio and write out the first four terms. { f n }={ 3 2n }In Problems 9-18, show that each sequence is geometric. Then find the common ratio and write out the first four terms. { t n }={ 3 n1 2n }In Problems 9-18, show that each sequence is geometric. Then find the common ratio and write out the first four terms. { u n }={ 2 n 3 n1 }In Problems 19-26, find the fifth term and the n th term of the geometric sequence whose initial term a 1 and common ratio r are given. a 1 =2;r=3In Problems 19-26, find the fifth term and the n th term of the geometric sequence whose initial term a 1 and common ratio r are given. a 1 =2;r=4In Problems 19-26, find the fifth term and the n th term of the geometric sequence whose initial term a 1 and common ratio r are given. a 1 =5;r=1In Problems 19-26, find the fifth term and the n th term of the geometric sequence whose initial term a 1 and common ratio r are given. a 1 =6;r=2In Problems 19-26, find the fifth term and the n th term of the geometric sequence whose initial term a 1 and common ratio r are given. a 1 =0;r= 1 2In Problems 19-26, find the fifth term and the n th term of the geometric sequence whose initial term a 1 and common ratio r are given. a 1 =1;r= 1 3In problems 1926, find the fifth term and the nth term of the geometric sequence whose first term a1 and common ratio r are given. a1=3;r=3In Problems 19-26, find the fifth term and the n th term of the geometric sequence whose initial term a 1 and common ratio r are given. a 1 =0;r= 1In Problems 27-32, find the indicated term of each geometric sequence. 7th terms of 1, 1 2 , 4 4 ,In Problems 27-32, find the indicated term of each geometric sequence. 8th terms of 1,3,9,In problems , find the indicated term of each geometric sequence.
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In Problems 27-32, find the indicated term of each geometric sequence. 10th terms of 1,2,4,In Problems 27-32, find the indicated term of each geometric sequence. 8th terms of 0.4,0.04,0.004,In Problems 27-32, find the indicated term of each geometric sequence. 8th terms of 0.1,1.0,10.0,In problems 3340, find the nth term an of each geometric sequence. When given, r is the common ratio. 6,18,54,162,.....In Problems 33-40, find the n th term a n of each geometric sequence. When given, r is the common ratio. 5,10,20,40,In Problems 33-40, find the n th term a n of each geometric sequence. When given, r is the common ratio. 3,1, 1 3 , 1 9 ,In Problems 33-40, find the n th term a n of each geometric sequence. When given, r is the common ratio. 4,1, 1 4 , 1 16 ,In Problems 33-40, find the n th term a n of each geometric sequence. When given, r is the common ratio. a 6 =243;r=3In Problems 33-40, find the n th term a n of each geometric sequence. When given, r is the common ratio. a 2 =7;r= 1 3In Problems 33-40, find the n th term a n of each geometric sequence. When given, r is the common ratio. a 2 =7; a 4 =1575In Problems 33-40, find the n th term a n of each geometric sequence. When given, r is the common ratio. a 3 = 1 3 ; a 6 = 1 81In problems 41-46, find each sum. 1 4 + 2 4 + 2 2 4 + 2 3 4 ++ 2 n1 4In problems 41-46, find each sum. 3 9 + 3 2 9 + 3 3 9 ++ 3 n 9In problems 41-46, find each sum. k=1 n ( 2 3 ) kIn problems 41-46, find each sum. k=1 n 4 3 k1In problems 41-46, find each sum. 1248( 2 n1 )In problems 41-46, find each sum. 2+ 6 5 + 18 25 ++2 ( 3 5 ) n147AYU48AYU49AYU50AYU51AYU52AYUIn Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 1+ 1 3 + 1 9 +In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 2+ 4 3 + 8 9 +In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 8+4+2+In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 6+2+ 2 3 +In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 2 1 2 + 1 8 1 32 +In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 1 3 4 + 9 16 27 64 +In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 8+12+18+27+In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 9+12+16+ 64 3 +In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. k=1 5 ( 1 4 ) k1In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. k=1 8 ( 1 3 ) k1In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. k=1 1 2 3 k1In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. k=1 3 ( 3 2 ) k1In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. k=1 6( 2 3 ) k1In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. k=1 4 ( 1 2 ) k1In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. k=1 3 ( 2 3 ) k68AYU69AYU70AYU71AYU72AYU73AYU74AYU75AYU76AYU77AYU78AYU79AYU80AYU81AYU82AYU83AYU84AYU85AYU86AYU87AYU88AYU89AYU90AYU91AYU92AYUSinking Fund Scott and Alice want to purchase a vacation home in 10 years and need 50000 for a down payment. How much should they place in a savings account each month if the per annum rate of return is assumed to be 3.5 compounded monthly?Sinking Fund For a child born in 2018, the cost of a 4- year college education at a public university is projected to be 185,000. Assuming a 4.75 per annum rate of return compounded monthly, how much must be contributed to a college fund every month to have 185,000 in 18 years when the child begins the college?95AYU96AYUMultiplier Suppose that, throughout the U.S. economy, individuals spend 90 of every additional dollar that they earn. Economists would say that an individual’s marginal propensity to consume is 0.90 . For example, if Jane earns an additional dollar, she will spend 0.9(1)=0.90 of it. The individual who earns 0.90 (from Jane) will spend 90 of it, or 0.81 . This process of spending continues and results in an infinite geometric series as follows: 1.0.90.0.90 2 , 0.90 3 .0.90 4 . The sum of this infinite geometric series is called the multiplier. What is the multiplier if individuals spend 90 of every additional dollar that they earn?Multiplier Refer to Problem 97. Suppose that the marginal propensity to consume throughout the U.S. economy is 0.95 . What is the multiplier for the U.S. economy?99AYU100AYU101AYU102AYU103AYU104AYU105AYU106AYU107AYU108AYU109AYUIn Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 2+4+6+...+2n=n( n+1 )In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 1+5+9+...+( 4n3 )=n( 2n1 )In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 3+4+5+...+( n+2 )= 1 2 n( n+5 )In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 3+5+7+...+( 2n+1 )=n( n+2 )In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 2+5+8+...+( 3n1 )= 1 2 n( 3n+1 )In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 1+4+7+...+( 3n2 )= 1 2 n( 3n1 )In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 1 +2+2 2 +... +2 n1 = 2 n 1In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 1 +3+3 2 +... +3 n1 = 1 2 ( 3 n 1 )In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 1 +4+4 2 +... +4 n1 = 1 3 ( 4 n 1 )In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 1 +5+5 2 +... +5 n1 = 1 4 ( 5 n 1 )In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 1 12 + 1 23 + 1 34 +...+ 1 n( n+1 ) = n n+1In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 1 13 + 1 35 + 1 57 +...+ 1 ( 2n1 )( 2n+1 ) = n 2n+1In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 1 2 + 2 2 + 3 2 +...+ n 2 = 1 6 n( n+1 )( 2n+1 )In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 1 3 + 2 3 + 3 3 +...+ n 3 = 1 4 n 2 ( n+1 ) 2In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 4+3+2+...+( 5n )= 1 2 n( 9n )In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 234...( n+1 )= 1 2 n( n+3 )In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 12+23+34+...+n( n+1 )= 1 3 n( n+1 )( n+2 )In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . 12+34+56+...+( 2n1 )( 2n )= 1 3 n( n+1 )( 4n1 )In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . n 2 +n is divisible by 2.In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n . n 3 +2n is divisible by 3.21AYU22AYU23AYU24AYU25AYU26AYU27AYU28AYU29AYU30AYU31AYUExtended Principle of Mathematical Induction The Extended Principle of Mathematical Induction states that if Conditions a and b hold, that is, a. A statement is true for a natural number j . b. If the statement is true for some natural number kj , then it is also true for the next natural number k+1 . Then the statement is true for all natural numbers j . Use the Extended Principle of Mathematical Induction to show that the number of diagonals in a convex polygon of n sides is 1 2 n( n3 ) . [Hint: Begin by showing that the result is true when n=4 (Condition (a).]Geometry Use the Extended Principle of Mathematical Induction to show that the sum of the interior angles of a convex polygon of n sides equals ( n2 ) 180 .34AYUThe ______ ______ is a triangular display of the binomial coefficients.2AYU3AYU4AYUIn Problems 5-16, evaluate each expression. ( 5 3 )6AYU7AYU8AYU