Question

Find the last term of the following sequence $x,1,\frac{1}{x}$ ........ to $30$ terms

Answer 1

It is given that the first term of G.P is $a_1=x$ and the second term is $a_2=1$, therefore,

$r=\frac{a_2}{a_1}=\frac{1}{x}$

We know that the formula for the $n$th term of an G.P is $T_n=a{r}^{n-1}$, where $a$ is the first term, $r$ is the common ratio.

It is given that the first term is $a=x$, number of terms are $n=30$ and the common ratio is $r=\frac{1}{x}$, thus,

$T_{30}=x\times {\left(\frac{1}{x}\right)}^{30-1}=x\times {\left(\frac{1}{x}\right)}^{29}=x\times \frac{1}{x^{29}}=\frac{1}{x^{28}}$

Hence, the last term is$\frac{1}{x^{28}}$.