Question

sa se arate ca 2(1+3+3la puterea 2+...+3la puterea8)<3 la puterea9

Answer 1

S = 2·(1+3+....+$3^{8}$) = 2+2·3+2·$3^{2} +...+ 3^{8}$ < 1+(2+2·3+2·$3^{2} +...+2$·$3^{8}$ )=3+2·3+2·$3^{2} +...+2$·$3^{8}$ (am adunat 1in fata)
Grapand termenii cate 2 se observa ca 3+2·3=3²
3²+2·3²=3³
.......samd
3$3^{n} +2$·$3^{n}$=$3^{n+1}$
Adica S<3+2·3+2·$3^{2} +...+2$·$3^{8}$=$3^{9}$