Question

8 la p 30 -7*8 la p 29 - ...- 7*8-1
Daca puteți sa îmi explicați cum se face.
PS-la p înseamnă la puterea

Answer 1

te follosesti de formula> $1+x+ x^{2} +....+ x^{n} = \frac{ x^{n+1} -1}{x-1}$
$8^{30} -7( 8^{29}+.....+8 )-1= \\ =8^{30} -7(1+8+.....+ 8^{29}-1 )-1 =\\ = 8^{30} -7 ( \frac{ 8^{29+1} -1}{8-1}-1 )-1= \\ = 8^{30} -7* \frac{ 8^{30} -1}{7} +7-1= \\ = 8^{30}- 8^{30}+1+7-1= \\ =7$

Answer 2

$8^k -7 \cdot 8^{k-1}=8^{k-1}(8-7)=8^{k-1}. \\ \\ Deci~S=(8^{30}-7 \cdot 8^{29})-7 \cdot 8^{28} - 7 \cdot 8^{27}-...-7 \cdot 8-1= \\ \\ ~~~~~~~~~~=(8^{29}-7 \cdot 8^{28})-7 \cdot 8^{27}-...-7 \cdot 8-1= \\ \\ ~~~~~~~~~~=8^{28}-7 \cdot 8^{27}-...-7 \cdot 8-1= \\ \\ ~~~~~~~~~~=............................= \\ \\ ~~~~~~~~~~=(8^2-7 \cdot 8)-1= \\ \\ ~~~~~~~~~~=8-1= \\ \\ ~~~~~~~~~~=7.$