Chapter 7.9, Problem 10CYU

To determine

To Find: The $nth$ term and seventh term of geometric sequence $-1,5,-\mathrm{25.............}$ .

Answer to Problem 10CYU

  $a_n=-{\left(-5\right)}^{n-1}\&a_7=-15625$

Explanation of Solution

Given information:

Geometric sequence is $-1,5,-\mathrm{25.............}$ .

Concept and Formula Used:

Geometric Sequence: Sequence in which each term after the first is found by multiplying the previous term by a constant called common ratio.

$nth$ term of the geometric sequence is $a_n=a{r}^{n-1}$ where ‘a’ is the first term and ‘r’ is the common ratio.

Calculation:

It is given that geometric sequence is $-1,5,-\mathrm{25.............}$

Common ratio is

  $\begin{array}{l}r=\frac{5}{-1}=\frac{-25}{5}=-5,a=-1\\ \text{Then }nth\text{ term of the sequence is}\\ a_n=\left(-1\right){\left(-5\right)}^{n-1}\\ a_n=-{\left(-5\right)}^{n-1}\end{array}$

Now, the seventh term is

  $\begin{array}{l}a=-1,r=-5\&n=7\\ a_7=-{\left(-5\right)}^{7-1}\\ a_7=-{\left(-5\right)}^6\\ a_7=-15625\end{array}$

Conclusion:

  $a_n=-{\left(-5\right)}^{n-1}\&a_7=-15625$