Question

Suppose Sn is defined as 2 + 22 + 23 + . . . + 2n . What is the next step in your proof of Sn = 2(2n - 1), after you verify that Sn is valid for n = 1?

A. Show that Sn is valid for n = k + 2.
B. Assume that Sn is valid for n = k .
C. Verify that Sn is valid for n = 1.
D. Show that Sn is valid for n = k.

Answer 1

Remark:


`S_n=2*1+2*2+2*3+...+2*n=2(1+2+3+...+n)`


`1+2+3+...+n= (n(n+1))/(2)`, by the famous Gauss formula.

So the formula for
`S_n` is:


`S_n=2*(n(n+1))/(2)=n(n+1)`



these types of formulas are proven by Induction.

The first step is proving for n=1,

then the next step is assuming Sn is valid for n=k.



Answer: B. Assume that Sn is valid for n = k .