Question

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

Answer #1

To calculate the sum we will use G.P.

3. Let N denote the nonnegative integers, and Z denote the
integers. Define the function g : N→Z defined by g(k) = k/2 for
even k and g(k) = −(k + 1)/2 for odd k. Prove that g is a
bijection.
(a) Prove that g is a function.
(b) Prove that g is an injection
. (c) Prove that g is a surjection.

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

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

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

Prove: If A is an n × n symmetric matrix all of whose
eigenvalues are nonnegative, then xTAx ≥ 0 for all
nonzero x in the vector space Rn.

Exercise 1. Prove that floor[n/2]ceiling[n/2] =
floor[n2/4], for all integers 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 induction prove that for all positive integers n, n^2−n is
even.

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