Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Precalculus Enhanced with Graphing Utilities
In 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 383AYU84AYU85AYU86AYU87AYU88AYU89AYU90AYU91AYU92AYU93AYU94AYU95AYU96AYU97AYU98AYU99AYU100AYU101AYU102AYU103AYU104AYU105AYU106AYU107AYUIn a(n) _________ sequence, the difference between successive terms is a constant.True or False For an arithmetic sequence { a n } whose first term is a 1 and whose common difference is d , the n th term is determined by the formula a n = a 1 +nd .If 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 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 =5;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 =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 25-30, find the indicated term in each arithmetic sequence. 90thtermof1,2,5,...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 39-56, find each sum. 5+9+13++49In Problems 39-56, find each sum. 2+5+8++41In Problems 39-56, find each sum. 73+78+83+88++558In 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++50In Problems 39-56, find each sum. n=1 80 ( 2n5 )In Problems 39-56, find each sum. n=1 90 ( 32n )In Problems 39-56, find each sum. n=1 100 ( 6 1 2 n )In Problems 39-56, find each sum. n=1 80 ( 1 3 n+ 1 2 )In Problems 39-56, find each sum. The sum of the first 120 terms of the sequence 14,16,18,20,...In Problems 39-56, find each sum. The sum of the first 46 terms of the sequence 2,1,4,7,...Find x so that x+3,2x+1,and5x+2 are consecutive terms of an arithmetic sequence.Find x so that 2x,3x+2,and5x+3 are consecutive terms of an arithmetic sequence.How many terms must be added in an arithmetic sequence whose first term is 11 and whose common difference is 3 to obtain a sum of 1092?How many terms must be added in an arithmetic sequence whose first term is 78 and whose common difference is 4 to obtain a sum of 702?Drury Lane Theater The Drury Lane Theater has 25 seats in the first row and 30 rows in all. Each successive row contains one additional seat. How many seats are in the theater?Football Stadium The corner section of a football stadium has 15 seats in the first row and 40 rows in all. Each successive row contains two additional seats. How many seats are in this section?Creating a Mosaic A mosaic is designed in the shape of an equilateral triangle, 20 feet on each side. Each tile in the mosaic is in the shape of an equilateral triangle, 12 inches to a side. The tiles are to alternate in color as shown in the illustration. How many tiles of each color will be required?Constructing a Brick Staircase A brick staircase has a total of 30 steps. The bottom step requires 100 bricks. Each successive step requires two fewer bricks than the prior step. (a) How many bricks are required for the top step? (b) How many bricks are required to build the staircase?Cooling Air As a parcel of air rises (for example, as it is pushed over a mountain), it cools at the dry adiabatic lapse rate of 5.5 F per 1000 feet until it reaches its dew point. If the ground temperature is 67 F , write a formula for the sequence of temperatures, { T n } , of a parcel of air that has risen n thousand feet. What is the temperature of a parcel of air if it has risen 5000 feet? Source: National Aeronautics and Space Administration64AYUSeats in an Amphitheater An outdoor amphitheater has 35 seats in the first row, 37 in the second row, 39 in the third row, and so on. There are 27 rows altogether. How many can the amphitheater seat?Stadium Construction How many rows are in the corner section of a stadium containing 2040 seats if the first row has 10 seats and each successive row has 4 additional seats?Salary If you take a job with a starting salary of 35,000 per year and a guaranteed raise of 1400 per year, how many years will it be before your aggregate salary is 280,000 ? [Hint: Remember that your aggregate salary after 2 years is 35,000+( 35,000+1400 ) .]Make up an arithmetic sequence. Give it to a friend and ask for its 20th term.Describe the similarities and differences between arithmetic sequences and linear functions.1AYU2AYU3AYU4AYU5AYU6AYU7AYU8AYU9AYU10AYU11AYU12AYU13AYU14AYU15AYU16AYU17AYU18AYU19AYU20AYU21AYU22AYU23AYU24AYU25AYU26AYU27AYU28AYU29AYU30AYU31AYU32AYU33AYU34AYU35AYU36AYU37AYU38AYU39AYU40AYU41AYU42AYU43AYU44AYU45AYU46AYU47AYU48AYU49AYU50AYU51AYU52AYU53AYU54AYU55AYU56AYU57AYU58AYU59AYU60AYU61AYU62AYU63AYU64AYU65AYU66AYU67AYU68AYU69AYU70AYU71AYU72AYU73AYU74AYU75AYU76AYU77AYU78AYU79AYU80AYU81AYU82AYU83AYU84AYU85AYU86AYU87AYU88AYU89AYU91AYU92AYU93AYU94AYU95AYU96AYU97AYU98AYU99AYU100AYU101AYUIn 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.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+2 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( n+1 )( n+2 ) is divisible by 6.In Problems 23-27, prove each statement. If x1 , then x n 1 .In Problems 23-27, prove each statement. If 0x1 , then 0 x n 1 .In Problems 23-27, prove each statement. ab is a factor of a n b n . [Hint: a k+1 b k+1 =a( a k b k )+ b k ( ab ) ]In Problems 23-27, prove each statement. a+b is a factor of a 2n+1 b 2n+1 .In Problems 23-27, prove each statement. ( 1+a ) n 1+na , for a0Show that the statement n 2 n+41 is a prime number� is true for n=1 but is not true for n=41 .Show that the formula 2+4+6++2n= n 2 +n+2 obeys Condition II of the Principle of Mathematical Induction. That is, show that if the formula is true for some k , it is also true for k+1 . Then show that the formula is false for n=1 (or for any other choice of n ).Use mathematical induction to prove that if r1 , then a+ar+a r 2 ++a r n1 =a 1 r n 1rUse mathematical induction to prove that a+( a+d )+( a+2d )++[ a+( n1 )d ]=na+d n( n1 ) 2Extended 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 .How would you explain the Principle of Mathematical Induction to a friend?The ______ ______ is a triangular display of the binomial coefficients.( n 0 )=and( n 1 )= .True or False ( n j )= j! ( nj )!n!The ______ ________ can be used to expand expressions like ( 2x+3 ) 6 .In Problems 5-16, evaluate each expression. ( 5 3 )In Problems 5-16, evaluate each expression. ( 7 3 )In Problems 5-16, evaluate each expression. ( 7 5 )In Problems 5-16, evaluate each expression. ( 9 7 )In Problems 5-16, evaluate each expression. ( 50 49 )In Problems 5-16, evaluate each expression. ( 100 98 )In Problems 5-16, evaluate each expression. ( 1000 1000 )In Problems 5-16, evaluate each expression. ( 1000 0 )In Problems 5-16, evaluate each expression. ( 55 23 )In Problems 5-16, evaluate each expression. ( 60 20 )In Problems 5-16, evaluate each expression. ( 47 25 )In Problems 5-16, evaluate each expression. ( 37 19 )In Problems 17-28, expand each expression using the Binomial Theorem. ( x+1 ) 5In Problems 17-28, expand each expression using the Binomial Theorem. ( x1 ) 5In Problems 17-28, expand each expression using the Binomial Theorem. ( x2 ) 6In Problems 17-28, expand each expression using the Binomial Theorem. ( x+3 ) 5In Problems 17-28, expand each expression using the Binomial Theorem. ( 3x+1 ) 4In Problems 17-28, expand each expression using the Binomial Theorem. ( 2x+3 ) 5In Problems 17-28, expand each expression using the Binomial Theorem. ( x 2 + y 2 ) 5In Problems 17-28, expand each expression using the Binomial Theorem. ( x 2 y 2 ) 6In Problems 17-28, expand each expression using the Binomial Theorem. ( x + 2 ) 6In Problems 17-28, expand each expression using the Binomial Theorem. ( x 3 ) 4In Problems 17-28, expand each expression using the Binomial Theorem. ( ax+by ) 5In Problems 17-28, expand each expression using the Binomial Theorem. ( axby ) 4In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x 6 in the expansion of ( x+3 ) 10In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x 3 in the expansion of ( x3 ) 10In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x 7 in the expansion of ( 2x1 ) 12In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x 3 in the expansion of ( 2x+1 ) 12In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x 7 in the expansion of ( 2x+3 ) 9In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x 2 in the expansion of ( 2x3 ) 9In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The 5th term in the expansion of ( x+3 ) 7In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The 3rd terms in the expansion of ( x3 ) 7In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The 3rd term in the expansion of ( 3x2 ) 9In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The 6th term in the expansion of ( 3x+2 ) 8In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x 0 in the expansion of ( x 2 + 1 x ) 12In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x 0 in the expansion of ( x 1 x 2 ) 9In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x 4 in the expansion of ( x 2 x ) 10In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x 2 in the expansion of ( x + 3 x ) 8Use the Binomial Theorem to find the numerical value of ( 1.001 ) 5 correct to five decimal places. [Hint: ( 1.001 ) 5 = ( 1+ 10 3 ) 5 ]Use the Binomial Theorem to find the numerical value of ( 0.998 ) 6 correct to five decimal places.Show that ( n n1 )=nand( n n )=1 .Show that if n and j arc integers with 0jn , then, ( n j )=( n nj ) Conclude that the Pascal triangle is symmetric with respect to a vertical line drawn from the topmost entry.If n is a positive integer, show that, ( n 0 )+( n 1 )++( n n )= 2 n [Hint: 2 n = ( 1+1 ) n ; now use the Binomial Theorem.]