Chapter 12, Problem 10CT

To determine

To find: The common ratio and the sum of the first $n$ terms.

Answer to Problem 10CT

The sum is $125\left[1-{\left(\frac{2}{5}\right)}^n\right]$ .

Explanation of Solution

Given information:

The given sequence is $25,10,4,\frac{8}{5},\mathrm{.......}$

Calculation:

Calculate the difference between the first and the second terms.

  $10-25=-15$

Calculate the difference between the second and the third terms.

  $4-10=-6$

Since the difference of the successive terms is not constant, we scan say that the given sequence is not arithmetic.

Calculate the ratio of the first and the second terms.

  $\begin{array}{c}r=\frac{10}{25}\\ =\frac{2}{5}\end{array}$

Calculate the ratio of the second and the third terms.

  $\begin{array}{c}r=\frac{4}{10}\\ =\frac{2}{5}\end{array}$

Since the ratio of the successive terms is a constant, we can say that the given sequence is with common ratio $\frac{2}{5}$ .

Substitute the values in above formula,

  $s_n=a_1\cdot \frac{1-r^n}{1-r}$ .

  $\begin{array}{c}{S}_n=25\left[\frac{1-{\left(\frac{2}{5}\right)}^n}{1-\frac{2}{5}}\right]\\ =125\left[1-{\left(\frac{2}{5}\right)}^n\right]\end{array}$

Therefore, the sum is $125\left[1-{\left(\frac{2}{5}\right)}^n\right]$