Question

Let $a _{ n }=\int\limits_{-1}^{ n }\left(1+\frac{ x }{2}+\frac{ x ^2}{2}+\frac{ x ^3}{3}+\ldots \ldots +\frac{ x ^{ n -1}}{ n }\right) dx$ for $n \in N$ Then the sum of all the elements of the set $\left\{ n \in N : a _{ n } \in(2,30)\right\}$ is ______

Answer 1

Correct Answer: 5

The correct answer is 5.

\(\int_{-1}^{n}(1+\frac{x}{2}+\frac{x^{2}}{3}+....+\frac{x^{n-1}}{n})dx\)

\([x+\frac{x^{2}}{2}+\frac{x^{3}}{3^{2}}+...\frac{x^{n}}{n^{2}}]^{n}\)

\((n+\frac{n^{2}}{2^{2}}+\frac{n^{3}}{3^{2}}+...+-\frac{n^{n}}{n^{2}})\)

\(-(-1+\frac{1}{2^{2}}-\frac{1}{3^{2}}+\frac{1}{4^{2}}+...+\frac{(-1)^{n}}{n^{2}})\)

\(a_{n}=(n+1)+\frac{1}{2^{2}}(n^{2}-1)+\frac{1}{3^{2}}(n^{3}+1)+...+\frac{1}{n^{2}}(n^{n}-(-1)^{n})\)

\(If \; n=1\Rightarrow a_{n}=2\not{\epsilon }(2,30)\)

If n = 2

\(\Rightarrow a_{n}=(2+1)+\frac{1}{2^{2}}(2^{2}-1)=3+\frac{3}{4}< 30\)

If n = 3

\(\Rightarrow a_{n}=(3+1)+\frac{1}{4}(8)+\frac{1}{9}(28)=11+\frac{28}{9}< 30\)

If n = 4

\(\Rightarrow a_{n}=(4+1)+\frac{1}{4}(16-1)+\frac{1}{9}(64+1)+\frac{1}{16}\)

\(=5+\frac{15}{4}+\frac{65}{9}+\frac{255}{16}> 30\)

Test {2,3} sum of elements 5