Can you assist  me in  verifing    that    for    all    n ≥ 1,    the    sum    of    the    first    2n positive    integers    is given    by    the     formula: 12 + 22 + 32 + ··· + (2n)2 = n(2n + 1)(4n + 1)                                                         3

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: Show that for every integer n, n³ + 2 is not divisible by 4.

A: Here, we use rules of divisibility.

Q: 1. The expression 16x? – 25y? is equivalent to: A) (4x – 5y)? C) (4x – 5y)(4x + 5y) B) (16 – 25)(x?…

A:

Q: Show that 7 divides 29(52k) + 17(25k-2) for any integer k ≥ 1.

A:

Q: Show that 3 · 4n + 51 is divisible by 3 and 9 for all positive integers n.

A:

Q: Find each (n+3)(n+2) (4n 5)(3n + 2) (n+7)(n + 5)

A: Since you have posted multiple subparts we solve the first three subparts for you to get the…

Q: ) Find an integer having the remainders 3, 11, 15 when divided by 10, 13, 17, respec- tively.…

A:

Q: If a number has n fraction (n>1), then the square of the number has 2n factors is this true

A:

Q: Add the polynomials (10z - 82 7æ³ - 10) + (- 4x + 12a + 7a + 9) | Submit Question

A: Add the polynomials. Solution is given below-

Q: let a=2+7i b=4-2i Choose the correct result of a/b*

A:

Q: Can someone explain how (n + 1) and (n + 2) can be integers? and also how can they be bigger than 1?

A: Answer: 1:  The solution is as given below:

Q: Expand the following binomials and simplify. a. (2x+ y)" b. (r? +3)* c. Gx+ 3y)*

A:

Q: Show that (a) 8n −3n is divisible by 5 for all natural numbers n.  (b) n2 −n +41 is odd for all…

A: expansion of (a+b)n = hco an bo + nc1 an-1 b1 +.......... +ncn ao b n

Q: Please help with both (a) and (b) in the picture.

A: Please refer the image below

Q: Find the product 3/2 (5 /12 + 6/6) and express your answer in the form A/B+ CVD, A, B, C, and D are…

A: Given : 32512 + 66

Q: Theorem: The sum of any even integer and any odd integer is odd. Construct a proof for the theorem…

A: We have to prove the theorem by scrambling the given statements.

Q: The number 42 has the prime factorization 2- 3-7. Thus, 42 can be written in four ways as a product…

A: Factorise  910 inti prine factors and the find the required pairs.

Q: Show that 3·4n+ 51 is divisible by 3 and 9 for all positive integersn

A: Consider the provided expression,

Q: zu +16n + 20.

A: Given: An expression n2+16n+20 To find All possible remainder when it is divided by 11.As per our…

Q: For each natural aumber a, the sum 1(2) + 2(3) ++ R(n + 1) is equal ta None of the choices This…

A:

Q: If k is a positive integer, what is the remainder when (k+ 2)(k3 - k) is divided by 6? Stilo…

A:

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: The product of any integer and 2 is greater than 2.

A:

Q: Show that 3 · 4n + 51 is divisible by 3 and 9 for all positive integers n.

A: Given , 3.4n+51 is divisible by 3 and 9 for all positive integers n''-1The statement 1 can be…

Q: um 1+4+7+..

A:

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: C. Find the possible values of a so that an + an+1 + an+2+ an+3 + an+4 will be divisible by 11.

A:

Q: 1. Expand and simplify each expression. a) (8x – 3) (6x + 7) b) 3(2x + 1) – 2(x -5)?

A:

Q: Simplify the following expressions by rewriting without any factorials. A. (n+1)! (n - 1)! (2n+1)!…

A: A)n2+n B)4n2+10n+61​ C) 4n2+2nExplanation:

Q: Please Prove whether it is True or False. The value of nª - n² is divisible by 6 for any integer…

A: To prove that whether the given statement is true or false