Prove 1+2+4+...+2^(n-1)=2^n - 1

Question

by mathematical induction?

Answer 1

The equation is 1+2+4+. . . .+2^(n-1) = 2ⁿ - 1.

Prove the statement for n = 1.

Left hand side (LHS) expression = 2^(n-1) = 2^(1-1) = 2⁰ = 1.

Right hand side (RHS) expression = 2ⁿ - 1 = 2¹ - 1 = 2 - 1 = 1.

Since LHS = RHS, for n = 1 the statement is true.

Assuming that the formula S_k = 1+2+4+. . . .+2^{(k -1)} = 2^k - 1 is true, we  must show that the formula S_k+1 = 2^{(k + 1)} - 1 is true.

S_k+1 = 1 + 2 + 4 +. . . .+ 2^{(k -1)} + 2^{[(k + 1) - 1]}.

S_k+1 = [1 + 2 + 4 +. . . .+ 2^{(k -1)}] + 2^k.

S_k+1 = [2^k - 1] + 2^k.    [Since 1+2+4+. . . .+2^(n-1) = 2ⁿ - 1]

S_k+1 = 2*2^k - 1.

S_k+1 = 2^{(k + 1)} - 1.