Question

prove by induction that (1/2) + (2/2^2) + (3/2^3)... + (n/2^n) = 2 - (n+2)/2^n​

Answer 1

The base case is when n=1: 2-3/2=1/2 according to the formula, and this is true because 1/2 is the first term.

Assume for S[n] that the formula is true. We know the nth term is n/(2^n) so we would expect:

S[n+1] to be S[n]+(n+1)/(2^(n+1)).

So, 2-(n+2)/(2^n)+(n+1)/(2^(n+1))=

2-2(n+2)/(2^(n+1))+(n+1)/(2^(n+1))=

2+(n+1-2n-4)/(2^(n+1))=

2+(-n-3)/(2^(n+1))=

2-((n+1)+2)/(2^(n+1))=S[n+1].

This proves by induction that the formula is correct.