To prove 3+7+11+...+(4n − 1) = n(2n + 1) holds for an arbitrary n € Z+, in the inductive step, under our previous assumption, we prove that Check ALL that apply, i.e. you are allowed to choose ONE or MORE answers. OP(n+1): 3+7+11+. . +4(n+1)-1= 2n² + 5n +31 P(k+1): 3+7+11+...+ 4(k+ 1) − 1 = (k+1)(2(k+1)+1) n+1 P(n+1): (4(+1) − 1) = (n + 1)(2(n + 1) + 1) j=1 n+1 P(n+1): (4j-1) = (n + 1)(2(n+1) + 1) Σw-

Q: 32-2n Σ=3 Ln=3_25-n

A: We have to evaluate ∑n=3∞ 32-2n25-n.

Q: n(n+1)(2n+1) Prove: 12 + 22 + 3² + ... +n? for all positive integers, n.

A:

Q: Find, with proof, all integer solutions to 15s + 27t = 6.

A:

Q: 3. Using induction, verify that 12 + 3 + 5² + (2n – 1)² = n(2n–1)(2n+1) is true for every positive…

A:

Q: 1+ x Σ Aa" (1 – 2)? n=0

A:

Q: If P (n) is the statement "n + n" is even", and if P(r) is true, then prove P (r + 1) is true.

A:

Q: 5. Find all integer solutions to 6s+ 14t+21u = 1.

A:

Q: Find all integer solutions to 25s+ 14t = 1. O

A: Given equation25s + 14t = 1We have to find all integer solutions for s and t.

Q: Prove by induction that (−2)º + (−2)¹+ (−2)² + ... + (−2)¹ : = for all n positive odd integers.…

A: Let's start by plugging in n=1 into the formula given:   (-2)^0 + (-2)^1 = (1 - 2^(1+1))/3   1 +…

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: For positive integers n and k (k sn+ 1), the kth term of (a + b)n is given by

A: The binomial expansion of (a+b)^n is

Q: use gauss's trick to determine R(n)=2+3+4+...+(n-1)+n+(n+1)

A: It is known that sum of first n numbers is n(n+1)2  that is  1+2+3+----+n=n(n+1)2 Gauss trick method…

Q: ○〇 k + 1 -1) 43k k=1

A:

Q: Hence show that for positive integers n and m: P = I P P² for n = 3m for n = = 3m + 1 for n = 3m +2

A:

Q: For each integer n2 2, let P(n) be the following equation (n – 1)n(2n + 5) (i – 1)(i + 1) = 72 - 6…

A:

Q: (Vn)(n2 +n+ 17 is prime)

A: Since, you have posted multiple questions , I am doing the very question (ques 7)  according to our…

Q: (2) Give a proof by induction of the formula: P(n): 12 +22 + ...+ n² = "(n+ 1)(2n + 1) 6

A:

Q: Select all of the following that are equivalent to >(2) +> (3k + 1)? k=0 k=0 п+1 A. O) (3k + 3) k=0…

A:

Q: 1. Use the principle of mathematical induction to show that n(3n-1) P(n) = 1+4 + 7 + 10 + ... + (3n…

A:

Q: 1. Prove: If n is odd then 5n + 4 is odd.

A:

Q: is a1 = 2, a2 =3, and an  = 5an–1 – 4an–2 for n > 2 expressed in explicit formula? true or false?

A:

Q: n (A)-0) k Simplify the equation shown here. with n≥k and n, ke Z+ (In the answer choices, nCk…

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: