Question

Chứng minh rằng : 3 + 3^2 + 3^3 + 3^4 + ... +3^100 chia hết cho 120. (gợi ý : nhóm thành 25 nhóm mỗi nhóm có 4 số hạng )

cho E = 1/3 + 2/3^2 + 3/3 ^3 + 4/3^4 + ... +100/3^100. chứng minh rằng E <3/4

giúp mình 2 bài này nhé

Answer 1

\(E=\frac{1}{3}+\frac{2}{3^2}+\frac{3}{3^3}+...+\frac{100}{3^{100}}\)

\(3E=1+\frac{2}{3}+\frac{3}{3^2}+...+\frac{100}{3^{99}}\)

\(3E-E=\left(1+\frac{2}{3}+\frac{3}{3^2}+...+\frac{100}{3^{99}}\right)-\left(\frac{1}{3}+\frac{2}{3^2}+\frac{3}{3^3}+...+\frac{100}{3^{100}}\right)\)

\(2E=1+\frac{1}{3}+\frac{1}{3^2}+....+\frac{1}{3^{99}}-\frac{100}{3^{100}}\)

\(6E=3+1+\frac{1}{3}+...+\frac{1}{3^{98}}-\frac{100}{3^{99}}\)

\(6E-2E=\left(3+1+\frac{1}{3}+...+\frac{1}{3^{98}}-\frac{100}{3^{99}}\right)-\left(1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{99}}-\frac{100}{3^{100}}\right)\)

\(4E=3-\frac{100}{3^{99}}-\frac{1}{3^{99}}+\frac{100}{3^{100}}\)

\(4E=3-\frac{300}{3^{100}}-\frac{3}{3^{100}}+\frac{100}{3^{100}}\)

\(4E=3-\frac{203}{3^{100}}< 3\)

\(\Rightarrow4E< 3\)

\(\Rightarrow E< \frac{3}{4}\left(đpcm\right)\)

Answer 2

Bài 1:

Ta có: \(3+3^2+3^3+...+3^{100}\)

\(=\left(3+3^2+3^3+3^4\right)+....+\left(3^{97}+3^{98}+3^{99}+3^{100}\right)\)

\(=120+3^5\left(3+3^2+3^3+3^4\right)+....+3^{96}\left(3+3^2+3^3+3^4\right)\)

\(=120+3^5.120+...+3^{96}.120\)

\(=120.\left(1+3^5+.....+3^{96}\right)\)

\(\Rightarrow3+3^2+3^3+3^4+....+3^{100}\)chia hết cho 120 (vì có chứa thừa số 120)