ʰᵉʳᵉ ⁱˢ ᵒⁿᵉ ᵖᵒˢˢⁱᵇˡᵉ ʷᵃʸ ᵒᶠ ᵗʰⁱⁿᵏⁱⁿᵍ ᵃᵇᵒᵘᵗ ᵖᵒʷᵉʳˢ ᵒᶠ ²:

March 20, 2026Mathematics

ScanSolve AI Answer

March 20, 2026

Answer

$2^{n} - 1$

Step 1: Let $S$ be the sum of the first $n$ powers of $2$ , starting from $2^0$ .

S = 1 + 2 + 4 + \dots + 2^{n-1}

Step 2: Multiply $S$ by $2$ to shift the terms.

2S = 2 + 4 + 8 + \dots + 2^n

Step 3: Subtract the original $S$ from $2S$ .

2S - S = (2 + 4 + 8 + \dots + 2^n) - (1 + 2 + 4 + \dots + 2^{n-1})

S = 2^n - 1

The sum is $\boxed{2^{n} - 1}$ .

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