Chapter 12.3, Problem 77AYU

To determine

To find: Whether the sequence is arithmetic, geometric or neither, find the common difference or common ratio and also find the sum of the first fifty terms if the sequence is arithmetic or geometric.

Answer to Problem 77AYU

The sequence is geometric, common ratio is $-2$ and the sum of first $50$ terms is $3.75299\times {10}^{14}$ .

Explanation of Solution

Given information:

The series is $-1,2,-4,\mathrm{8.....}$ .

Calculation:

Here, $a_1=-1$ , $a_2=2$ and $a_3=-4$ .

Calculate the common ratio.

  $\begin{array}{c}r=\frac{a{2}_{}}{a_1}\\ =\frac{2}{-1}\\ =-2\end{array}$

Calculate the sum of first $n$ terms.

  $\begin{array}{c}{S}_n=a_1\frac{r^n-1}{r-1}\\ =\left(-1\right)\frac{{\left(-2\right)}^n-1}{-2-1}\\ =\frac{1}{3}\left\{{\left(-2\right)}^n-1\right\}\end{array}$

Put $n=50$ and calculate the sum of first $50$ terms from above equation.

  $\begin{array}{c}{S}_{50}=\frac{1}{3}\left\{{\left(-2\right)}^n-1\right\}\\ =\frac{1}{3}\left\{{\left(-2\right)}^{50}-1\right\}\\ =3.75299\times {10}^{14}\end{array}$

Therefore, the sequence is geometric, common ratio is $-2$ and the sum of first $50$ terms is $3.75299\times {10}^{14}$ .