Problems: Use mathematical induction to prove the following statements. Note: In

Question

Problems: Use mathematical induction to prove the following statements. Note: In the inductive step, be sure to start with the inductive hypothesis and deduce P(k+1) from it.

1. For every positive integer n, the following formula is true: 4*(5^1)+4*(5^2)+...+4*(5^n)=[5^(n+1)]-5.

2. For every positive integer n, the following formula is true: 1/(1*2)+1/(2*3)+...+1/[n(n+1)]=n/(n+1).

3. For every positive integer n, the following matrix formula is true:
|1 2|^n
|0 2|
=
|1 2^n+1-2|
|0 2^n |.

4. For every positive integer n, the inequality n!? n^n is true.


How do do this using the basis step, inductive step, and inductive hypothesis in the proof.

Explanation / Answer

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