By considering the summation: 2+22+23+…..+2n-1
A). What is the formula used for the summation? ________________________
B). Prove that your formula is correct for all integers n>=2 by mathematical induction.
A) For every natural number n, 21 + 22 + 23... + 2n-1 = 2n − 2 (Lets call this equation P(n))
B) Proof:
Let n = 2, then
LHS
=====
= 2
RHS
=========
= 22 - 2 = 4 - 2= 2
Let us assume P(k) is true for n = k.
We need to prove that the equation P(k+1) is true for n = k+1
For P(k+1):
LHS
===========
= 21 + 22 + 23 + .. + 2k-1 + 2k = (21 + 22 + 23 + .. + 2k-1) + 2k
= 2k− 2 + 2n
= 2.2k - 2 = 2k+1 - 2 (proved)
By considering the summation: 2+22+23+…..+2n-1 A). What is the formula used for the summation? ________________________ B)....
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