Prove Statement using the First Principle of Mathematical Induction

Question

Prove the following statements using the First Principle of Mathematical Induction:

(a) If r ≠ 1, then

1 + r + r² + ... + rⁿ = (1 − rⁿ⁺¹)/(1-r)

for all n ∈ N. Why does the formula not work when r = 1?, what will be the “formula” for 1 + r + r² + ... + rⁿ when r = 1?

(b) For all n ∈ N with n ≥ 2,

3ⁿ > 3n + 1.

Is the previous result true for n = 1?

(c) 5 | (4²ⁿ − 1) for all n ∈ N.

Answer 1

For induction you have 3 steps:

 

(1) The basis step: Prove P(0) true      (or in some cases P(1))

(2) Assume P(k) is true

(3) Prove P(k+1) is true

 

Let P(n) be : 1 + r + r² + ... + rⁿ = (1 − rⁿ⁺¹)/(1-r)

 

(1) When n=0 P(0) is just  1 = (1-r)/(1-r)

 

(2) Assume  1 + r + r² + ... + r^k = (1 − r^k+1)/(1-r) is true

 

(3) Now prove P(k+1):  1 + r + r² + ... + r^k + r^k+1 = (1 − r^k+2)/(1-r)

 

Use algebra (and some thought) to use (2) to get to (3).

 

 

This 3-step technique should work for b and c.

 

Any questions? Comment back.

 

 

Answer 2

(a)

For r = 1,  (1 − rⁿ⁺¹)/(1-r) = 0 / 0

 

(b) 

For n = 1, 3¹ ≠ 3 + 1

 

(c)

What is the question?