Question

Show by induction that 1+3+5+...+(2n-1) = n^2 for all n in the set of Natural Numbers

Answer #1

Prove the following using induction:
(a) For all natural numbers n>2, 2n>2n+1
(b) For all positive integersn,
1^3+3^3+5^3+···+(2^n−1)^3=n^2(2n^2−1)
(c) For all positive natural numbers n,5/4·8^n+3^(3n−1) is
divisible by 19

prove that 2^2n-1 is divisible by 3 for all natural numbers n ..
please show in detail trying to learn.

(a) use mathematical induction to show that 1 + 3 +.....+(2n +
1) = (n + 1)^2 for all n e N,n>1.(b) n<2^n for all n,n is
greater or equels to 1

Prove by induction on n that 13 | 2^4n+2 + 3^n+2 for all natural
numbers n.

Used induction to proof that 1 + 2 + 3 + ... + 2n = n(2n+1) when
n is a positive integer.

. Prove that 2^(2n-1) + 3^(2n-1) is divisible by 5 for
every natural number n.

Find all natural numbers n so that
n3 + (n + 1)3 > (n +
2)3.
Prove your result using induction.

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

Show that the number of labelled simple graphs with n vertices
is 2n(n-1)/2. (By Induction)

Exercise 6.6. Let the inductive set be equal to all natural
numbers, N. Prove the following propositions. (a) ∀n, 2n ≥ 1 +
n.
(b) ∀n, 4n − 1 is divisible by 3.
(c) ∀n, 3n ≥ 1 + 2 n.
(d) ∀n, 21 + 2 2 + ⋯ + 2 n = 2 n+1 − 2.

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