Відповіді на вправи: 703

703. Знайдіть суму:

1) 1 – 2 + 4 – … + 256.

b₁ = 1; b₂ = –2; q = $\frac{-2}{1}$ = –2.

b_n = b₁q^n–1;

1(–2)ⁿ • (–2)^–1 = 256; (–2)ⁿ(–$\frac{1}{2}$) = 256;

(–2)ⁿ = –512; (–2)ⁿ = (–2)⁹, n = 9.

S₉ = $\frac{1•\left(\right(-2{)}^9-1)}{-2-1}$ = $\frac{-513}{-3}$ = 171.

2) $\frac{1}{4}$ + $\frac{1}{8}$ + $\frac{1}{10}$ + … + $\frac{1}{512}$.

b₁ = $\frac{1}{4}$; b₂ = $\frac{1}{8}$; q = $\frac{1}{8}$ • $\frac{4}{1}$ = $\frac{1}{2}$.

b_n = b₁q^n–1;

$\frac{1}{512}$ = $\frac{1}{4}$($\frac{1}{2}$)^n–1; ($\frac{1}{2}$)ⁿ = $\frac{1}{512}$ • $\frac{2}{1}$;

($\frac{1}{2}$)ⁿ = $\frac{1}{256}$; ($\frac{1}{2}$)ⁿ = ($\frac{1}{2}$)⁸, n = 8.

S₈ = $\frac{\frac{1}{4}\left(\right(\frac{1}{2}{)}^8-1)}{\frac{1}{2}-1}$ = $\frac{127}{512}$.