(a) Prove that for all integer n 2 1, 3|n(n² + 11).

Q: 2.) Let 1 > -1 be a real number. Prove that (1+1)" >1+ nx for all integers n > 1.

A: We use principal of mathematical induction (PMI) to prove this statement.

Q: Prove that for all positive integers n, we have 6″ – (-4)″ Σ (1) 5k 2 k−1 k odd =

A: Expand 1+xn 1+xn=C0n(1)nx0+C1n(1)n-1x1+C2n(1)n-2x2+.......+Cnn(1)0xn=∑r=0nCrnxn Expand 1-xn…

Q: Prove that the statement is true for every positive integer n.

A: -11+-12+-13+---+-1n=-1n-12. We prove this statement by using the Principle of Mathematical…

Q: Prove by the principle of mathematical induction that 1 x 1!+2 x2! + 3 x 3! +…+ n x n!= (n+1)!—1…

A: to prove the result 1×1!+2×2!+3×3!+...+n×n!=n+1!-1 for all natural numbers n , proceed as follows…

Q: Let n be a positive integer. Prove that 1:(1)+2 (;) . (:)- n n + 2 . +n . n2"-1. ... n

A: We have to prove the identity: 1·n1+2·n2+....+n·nn=n2n-1

Q: Prove that the statement is true for every positive integer n.

A: Prove that the statement is true for every positive integer n.

Q: Prove that for all positive integers, 1 1 + 3.4 1 ... 1.2 2-3 n(n + 1) n+1'

A:

Q: Prove that the statement is true for every positive integer n.n < 2n

A: n<2n Let us prove this statement by using the Principle of Mathematical Induction. When n=1,…

Q: 2. Let an = 3n + 2 for n = 0, 1, 2, ... (i ) Find ao, a1 and a2. (ii ) Show that a2 = 4a1 – 3ao.…

A: Answer and explanation is given below..

Q: Prove that the statement is true for every positive integer n.

A: To prove: 1+4+7+...+3n-2=n3n-12

Q: Use induction to prove (2k – 1) = n² – 4 k=3 for all positive integers n2 3.

A:

Q: Prove that the statement is true for every positive integer n.

A: Given that, 2 is a factor of n2+n   We have to prove this statement is true for every positive…

Q: Prove that the statement is true for every positive integer n.

A: Given arithmetic progression (AP) is: 2+5+8+.........+3n-1 Here the first term is 2 and the common…

Q: 2n 10. Prove that 31 (5 - 1) for every integer n 2 0.

A:

Q: 2n+1+(-1)" Prove that for all integers n 2 0, 3.2"

A:

Q: n Σ k (k!) = (n+ 1)! - 1. %3D k=1

A:

Q: Prove by mathematical induction that for all positive integer values of n: n a 2 (r3 – r) = }(n – 1)…

A: As per our company policy we are supposed to answer only the first question. Kindly re Post other…

Q: Prove that n²– 3n + 1 is odd for every integer n.

A: To prove that n^2-3n+1 is odd for every integer n.

Q: prove that (1- (1/4))(1-(1/9))(1-(1/16))...(1-(1/ n2)) =(n+1)/2n for all integers n greater than or…

A: We have to prove that  (1- (1/4))(1-(1/9))(1-(1/16))...(1-(1/ n2)) =(n+1)/2 By mathematical…

Q: Find all integers n such that n| (2n+3). *

A: Given, n|(2n+3)     TO find, All integers n

Q: true or false In the domain of Gaussian integers Z[i], the element 17 is Irreducible

A:

Q: Use induction to prove that n п(п+1) Σ- k for all n e N. 2 k=0

A:

Q: Let a = 1, and let an+1 4a, – 1, for all nonnegative integers n. Prove that an 2-1"+1 3 %3D

A:

Q: Prove that the statements here are true for every positive integer n.   1 · 2 + 2 · 3 + 3 · 4 + · ·…

A: To prove: For every positive integer n, 1.2+2.3+3.4+.......+n(n+1)=n(n+1)(n+2)3

Q: Let k be some positive integer. Prove that a number of the form 4k +3 can never be the sum of two…

A:

Q: r(r+1) n+1 8. Prove by mathematical induction that 5"-4n - 1 is divisible by 16, for all positive…

A: As per our guidelines we are supposed to solve only one answer . Please repost another question as…

Q: (e) Prove that for any non-negative integer n, i 3 n - 3"+1 – (n+ 1)3" + 1] 4 j=0 (f) Prove that for…

A: (e)…

Q: Use the factor theorem to verify the statement

A: Consider the statement  x−y is a factor of xn−yn for every positive integer n.

Q: Prove that there exists an n ∈ N such that 7 | (3^n − 1).

A:

Q: 1. Prove by induction that for all positive integers n, we have: 1+3+5+ + (2n 1) n2

A:

Q: Prove that for all integers n, n2 – n + 3 is odd.

A: Suppose n is any integer .By the quotient remender theorem with n=2 ,n is either even or odd. Case…

Q: Prove that gn = 5 · 3n + 7 · 2n for all integers n ≥ 0.

A:

Q: a) Prove that ε n(n+ 1)(2n + 1) Σ 6 i=0

A:

Q: 6. Prove that for all n E w, the equality 0 n = 0 holds.

A: We will prove by induction. We need to prove for all n∈w, the equality 0·n=0 holds Let n=1, then,…

Q: (iii) Prove that there is a constant C such that 0(2º) 1.

A:

Q: Prove that n2 + 4 is not divisible by 3 for all integers n.

A:

Q: n Prove that (3i – 2) = ; (3n – 1) for all integers n > 1. - i=1

A:

Q: Prove that the statement is true for every positive integer n.

A: Given, Prove this statement is true for every positive integer n: 2+4+6+..+2n=n(n+1)

Q: 2""(-1)" +1 2. Prove that 1-2+2² – 2³ +...+(-1)" 2" =- for all positive integers n.

A: We prove this result by using mathematical induction. For answer please look at step 2.

Q: Prove that (f^2)_1 + (f^2)_2 + ⋯ + (f^2)_n = f_n*f_(n+1) when n is a positive integer.

A: We have to show that  f12+f22+f32+.....fn2=fnfn+1 where n is a positive integer and fn denotes the…

Q: 18. For every positive integer n, 2n < 2".

A:

Q: Q.No.6. Prove that by mathematically Induction 2" < (n + 1)!, for all integers n2 2.

A:

Q: A. Prove that n3 + 2n is divisible by 3 for all integers n.

A:

Q: 1.Then for positive integer n, A" is 0 1 Let A = 0 n² In 1 II In 0 1 II 0 0 2n In-1 n-1 2.

A:

Q: 3n Prove that for every integer n > 0, 9|(4" + 8).

A: To prove: For every integer n≥0, 9|43n+8. Proof by induction method.

Q: Prove that n2 − 3n + 1 is odd for every integer n.

A: Prove the result using mathematical induction. P(n):n2-3n+1. Check whether the result is true for n…

Q: Prove that for all positive integers n, we have 6″ Σ (1) * _ "(-¹)" 2 k=1 k odd IG

A:

Q: In an inductive proof that for every positive integer n, n п(п + 1)(2n + 1) Σ 6. j=1 what must be…

A: