Question

Prove that for all positive integers n,

(1^3) + (2^3) + ... + (n^3) = (1+2+...+n)^2

Answer #1

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Problem 1. Prove that for all positive integers n, we have 1 + 3
+ . . . + (2n − 1) = n ^2 .

Prove that 1−2+ 2^2 −2^3 +···+(−1)^n 2^n =2^n+1(−1)^n+1 for all
nonnegative integers n.

Use Mathematical Induction to prove that 3 | (n^3 + 2n) for all
integers n = 0, 1, 2, ....

Find all integers n (positive, negative, or zero) so
that (n^2)+1 is divisible by n+1.
ANS: n = -3, -2, 0, 1

Exercise 1. Prove that floor[n/2]ceiling[n/2] =
floor[n2/4], for all integers n.

Prove Euler’s theorem: if n and a are positive integers with
gcd(a,n)=1, then aφ(n)≡1 modn, where φ(n) is the Euler’s function
of n.

Statement: "For all integers n, if n2 is odd then n is odd"
(1) prove the statement using Proof by Contradiction
(2) prove the statement using Proof by Contraposition

Prove that n − 1 and 2n − 1 are relatively prime, for all
integers n > 1.

.Prove that for all integers n > 4, if n is a perfect square,
then n−1 is not prime.

Using PMI prove that the sum of the first n positive odd
integers is n2?
Is there a way to prove it substituting n+1 for n in the LHS and
RHS?

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