Hello, Neffets!

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Show by induction: .2 + 2^2 + 2^3 + . . . + 2^n .= .2(2^1 - 1}

Verify S(1): .2 .= .2(2^1 - 1) . . . yes!

Assume S(k): .2 + 2^2 + 2^3 + . . . + 2^k .= .2(2^k - 1)

Add 2^{k+1} to both sides:

. . 2 + 2^2 + 2^3 + . . . + 2^k + 2^{k+1} .= .2(2^k - 1) + 2^{k+1}

The right side is: .2^{k+1} - 2 + 2^{k+1} .= .2·2^{k+1} - 2 .= .2(2^{k+1} - 1)

And we have: .2 + 2^2 + 2^3 + . . . + 2^{k+1} .= .2(2^{k+1} - 1)

. .There!