Use mathematical induction to prove

Question

Use mathematical induction to prove 2¹ + 2² + 2³ + ... +2ⁿ = 2ⁿ⁺¹ - 2 is true for all positive integer of n

Answer 1

if n=1, 2 = 2²-2. verified for initial case.

Now assume that 2¹ + 2² +  ... +2^k = 2^k+1 - 2 IS true for some particular value, k.

 

Assuming the above, we add 2^k+1 to each side, and get

2¹ + 2² +  ... +2ⁿ + 2^k+1= 2^k+1 - 2 + 2^k+1

                                      = 2^{k+2 -2}

and now we see that THIS equation represents the case where n=k+1.

Therefore the truth of case n=k+1 is simply based on the truth of its predecessor case, n=k.

 

Since the case when k=1 is true, it follows (as an algebraic identity) that the case k+1 follows, so it's true when k=2.  Repeated reasoning tells us that case n=2 (true) dictates, therefore, the truth of case n=3, et cetera, et cetera.

 

Proved by the principle of Mathematical Induction.