Question

a) (-2)^1+(-2)^2+(-2)^3+...+(-2)^100=
b) (-3)^1+(-3)^2+(-3)^3+...+(-3)^75=​

Answer 1

Explicație pas cu pas:

a)

$a = {( - 2)}^{1} + {( - 2)}^{2} + {( - 2)}^{3} + ... + {( - 2)}^{100} = - {2}^{1} + {2}^{2} - {2}^{3} + {2}^{4} - {2}^{5} + {2}^{6} - ... - {2}^{97} + {2}^{98} - {2}^{99} + {2}^{100} = - ({2}^{1} + {2}^{3} + {2}^{5} + ... + {2}^{97} + {2}^{99}) + ({2}^{2} + {2}^{4} + {2}^{6} + ... + {2}^{98} + {2}^{100} ) = - 2(1 + {2}^{2} + {2}^{4} + ... + {2}^{96} + {2}^{98}) + {2}^{2}(1 + {2}^{2} + {2}^{4} + ... + {2}^{96} + {2}^{98}) = - 2S + 4S = 2S$

$S = 1 + {2}^{2} + {2}^{4} + ... + {2}^{96} + {2}^{98} = \\ = 1 + {4}^{1} + {4}^{2} + ... + {4}^{48} + {4}^{49}$

$4S = {4}^{1} + {4}^{2} + ... + {4}^{49} + {4}^{50}$

$4S + 1 = 1 + {4}^{1} + {4}^{2} + ... + {4}^{49} + {4}^{50}$

$4S + 1 = S + {4}^{50}$

$3S = {4}^{50} - 1 \implies S = \frac{{4}^{50} - 1}{3} \\ \implies \bf a = \frac{2({4}^{50} - 1)}{3} \iff a = \frac{2({2}^{100} - 1)}{3}$

b)

$b = {( - 3)}^{1} + {( - 3)}^{2} + {( - 3)}^{3} + ... + {( - 3)}^{75} = - {3}^{1} + {3}^{2} - {3}^{3} + {3}^{4} - {3}^{5} + {3}^{6} - ... - {3}^{71} + {3}^{72} - {3}^{73} + {3}^{74} - {3}^{75} = - ({3}^{1} + {3}^{3} + {3}^{5} + ... + {3}^{71} + {3}^{73}) + ({3}^{2} + {3}^{4} + {3}^{6} + ... + {3}^{72} + {3}^{74} ) - {3}^{75} = - 3(1 + {3}^{2} + {3}^{4} + ... + {3}^{70} + {3}^{72}) + {3}^{2}(1 + {3}^{2} + {3}^{4} + ... + {3}^{70} + {3}^{72}) - {3}^{75} = - 3S + 9S - {3}^{75} = 6S - {3}^{75}$

$S = 1 + {3}^{2} + {3}^{4} + ... + {3}^{70} + {3}^{72} = \\ = 1 + {9}^{1} + {9}^{2} + ... + {9}^{35} + {9}^{36} \\ 9S = {9}^{1} + {9}^{2} + ... + {9}^{35} + {9}^{36} + {9}^{37} \\ 9S + 1 = 1 + {9}^{1} + {9}^{2} + ... + {9}^{35} + {9}^{36} + {9}^{37} \\ 9S + 1 = S + {9}^{37} \iff 8S = {9}^{37} - 1 \\ S = \frac{{9}^{37} - 1}{8} \iff S = \frac{{3}^{74} - 1}{8}$

$b = \frac{6({3}^{74} - 1)}{8} - {3}^{75} = \frac{3({3}^{74} - 1)}{4} - {3}^{75} = \frac{{3}^{75} - 3 - 4 \cdot {3}^{75}}{4} = \frac{ - 3 \cdot {3}^{75} - 3}{4}$

$\implies \bf b = - \frac{3({3}^{75} + 1)}{4}$