Question

a).1+2+2^2+2^4+....+2^70=2^71-1
b). 1+3+3^2+.....+3^99=(3^100-1):2
c). 5+5^2+5^3+.....+5^24=[5*(5^24-1):4​

Answer 1

Explicație pas cu pas:

a)

$S = 1 + 2 + 2^{2} + 2^{3} + ... + 2^{70} \ \ \Big| \cdot 2$

$2S = 2 + 2^{2} + 2^{3} + 2^{4} + ... + 2^{70} + 2^{71} \ \ \Big| + 1$

$2S + 1 = \underbrace{1 + 2 + 2^{2} + 2^{3} + 2^{4} + ... + 2^{70}}_{S} + 2^{71}$

$2S + 1 = S + {2}^{71} \implies S = {2}^{71} - 1$

b)

$S = 1 + 3 + 3^{2} + 3^{3} + ... + 3^{99} \ \ \Big| \cdot 3$

$3S = 3 + 3^{2} + 3^{3} + 3^{4} + ... + + 3^{99} + 3^{100} \ \ \Big| + 1$

$3S + 1 = \underbrace{1 + 3 + 3^{2} + 3^{3} + ... + 3^{99}}_{S} + 3^{100}$

$3S + 1 = S + {3}^{100} \iff 2S = {3}^{100} - 1$

$\implies S = \Big({3}^{100} - 1\Big) : 2$

c)

$S = 5 + 5^{2} + 5^{3} + ... + 5^{24} \ \ \Big| \cdot 5$

$5S = 5^{2} + 5^{3} + 5^{4} + ... + + 5^{24} + 5^{25} \ \ \Big| + 5$

$5S + 5 = \underbrace{5 + 5^{2} + 5^{3} + ... + 5^{24}}_{S} + 5^{25}$

$5S + 5 = S + {5}^{25} \iff 4S = {5}^{25} - 5$

$4S = 5 \cdot ({5}^{24} - 1) \\ \implies S = \Big[5 \cdot ({5}^{24} - 1\Big)\Big] : 4$