Prove (n + 6)² – (n + 2)² is always a multiple of 8. (Use direct proof.)

Q: If n be positive integer, show that the algebraic sum of the numerical coefficients in the expansion…

A: As given above that n be a positive integer. And the given expression is: 3x-2yn We have to prove…

Q: Given that the expression (n²+n+1) n! can be written as a(n + 2)! + b(n + 1)! + cn! where a, b, c…

A: The main objective is to find the sum value.

Q: (n+1)! (n-1)! = 10n

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Q: 2(5) – 3(1 – 6) 1. Is even integer? 5 2. List all even integers between -6 and 6. 3. Is 124/6…

A: Four questions are given. The solution to the four questions are given below.

Q: ind witnesses to show that n^3 − 5n^2 − 4n is O(n^3).

A: In this question, order of polynomial can be found with the help of Big-O notation. Big-O Notation…

Q: Show that 2 x ²dx (x²+1)²(x²2x+2) S 81 Z 775 50

A: FORMULAS:1.If f(x)=p(x)q(x) and degree of q(x) is greater than p(x) by atleast 2 and q(x) have only…

Q: (6.1) Prove if r is a positive, even integer then 10* – 1 is divisible by 11. (6.2) Prove if r is a…

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Q: 21 divides 4(n+1) + 5(2n-1) whenever n is a positive integer.

A: Steps and explanations are as follows:In case of any doubt, please let me know. Thank you.

Q: Evaluate ak+2 if ap

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Q: 3) (n² + 10n+18) ÷ (n+ 5)

A: To divide the following expression.n2+10n+18÷n+5

Q: 15- (x + 5)lnxdx =? a) (-21/4) - 12In4 b)O (21/4) - 12ln2 (23/4) - 12ln4 d) O (-23/4) + 12ln2 e)…

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Q: If x is positive, the first negative term in the expansion of (1 + x)27/5 is (a) 5th term (b) 6th…

A: Introduction :

Q: In(+2)-3

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Q: Prove by using proof by contradiction: If (3n)^2 is even, then n is even.

A: Any odd integer multiply by an odd integer is odd. Any even integer multiply by an even integer is…

Q: n (x - 1)" The sixth term of the expansion of x is 1287 8 X 3125 . Determine n.

A: Let us consider the expression a+bn. The binomial expansion of the given expression is , a+bn=C0n…

Q: This will be (n+k-1)= 29+6-1 29 -¹1) = 34 29 = 278, 256.

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Q: 5. (CLO3) Prove that (n+3) = ("+²) + (+¹) + () for every positive integer n.

A: We need to prove        n+3C2 = n+2C2 + n+1C1 + nC0    We know, for any positive integer n and k…

Q: 00 (2n +1)" n2n n=1

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Q: (Зп+5 Find infimum of A |n E N} n+9

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Q: given 7(x-1)=2(3x+2) prove x=11

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Q: How did you found the bounds 2 and 8

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Q: 8 105 = 8 8 8+102 +. + %3D (1+r)' (1+r)² (1+r)³ ' (1+r)*

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Q: Differentiate the following: (a) (2x – 1)*. (b)(5x + 1)* (3 – 2x)* -

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Q: Prove Vn € N, 1 1 + 1 2 + 1 3 + + 11 2n - 1 + 2n ≥1+ n 2

A: Let's prove the given inequality step by step:Base Case (n=1):11≥1+12This is true, as 11=1 and…

Q: 7. For n ≥ 1, verify that 12+32 +52 ++ (2n- +3²+5² · + 2n-1)² = (2n+1) 3

A: Ans : page 1:Ans page 2:

Q: >(6k – 4)(7k + 3) - k=1 Enter your answer as a number without any arithmetic symbols. Sum =

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Q: Deduce from Q2 that if an integer N can be written as N = x² + ny² , and if a prime number q | N can…

A: We will use standard division algorithms to deduce the given statement.

Q: * (51) + (1- 31) (Ada) (5-si) (-1+si) (multi ply) (multiply)

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Q: True or False: The summation 18+28+...+n8 is a polynomial of n of degree 8.

A: Given that the summation 18+28+...+n8. The objective is to check the statement " the summation…

Q: I also found that the expression was divisible by 13 at n=6 and n=10. Is there anyway you could…

A: Any number n can be written in the form          n = 3m, or 3m+1, or 3m + 2 for some integer m.…

Q: What is the coefficent of 5 in (x + 1)7 - x²(x+3) 5

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Q: I simplified 6. 3 - x + 5 7 1 + x - 4 3 – 6(x – 4) by multiplying by 1 and obtained - 1 + 7(x + 5)

A: We have, 3-6x+51+7x-4

Q: rel= = 0A 15, x² = 4(4x² - 1) If, 2 Prove that, x = √√5 +√√3

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Q: 2(5) – 3(1 – 6) Is even integer?

A: If an integer is divisible by 2, then it is called an even integer. For example, 2,4,6 etc. are even…

Q: Show that n! > 3ª for n > 7.

A: The given inequality is, n!>3n We need to prove that, the inequality is true for n≥7 Firstly,…

Q: If n = 12 m (me N), prove that "C₂ (2+√3)² "Co- - + "C4 (2+√√3)4 n "C6 (2+√3)6 ·+···=(-

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