Chapter 11, Problem 31CCSR

Solution

To calculate: The sum of the first eight terms of the given series.

The given series is a geometric series and the sum of the first eight terms is 11111.111

Given:The series $10000,\text{ 1000, }100,10,\mathrm{..}$

Concept used:Each term in an arithmetic series changes by a constant, that is a constant is added or subtracted to each term to get the next term.

Each term in a geometric series is obtained by multiplying or dividing a constant. There is a constant ratio r between two successive terms of a geometric series. The sum of the first n terms is calculated using the formula

  $S_n=\frac{a_1(1-r^n)}{1-r}$

where $a_1$ is first term of series

r is the common ratio

n is number of terms

Calculation:

Given series $10000,\text{ 1000, }100,10,\mathrm{..}$ .

The series has a constant ratio dividing successive terms, as shown below

  $\frac{1000}{10000}=\frac{100}{1000}=\frac{10}{100}=\frac{1}{10}$

Thus the given series is a geometric series with $r=\frac{1}{10}$

The sum of first eight terms is calculated as shown below

  $\begin{array}{l}{S}_n=\frac{a_1(1-r^n)}{1-r}\\ \text{First term }{a}_1=10000\\ \text{Common ratio }r=\frac{1}{10}\\ \text{number of terms }n=8\end{array}$

  $\begin{array}{l}{S}_8=\frac{10000\left(1-{\left(\frac{1}{10}\right)}^8\right)}{1-\left(\frac{1}{10}\right)}\\ S_8=\frac{10000\left(1-\frac{1}{100000000}\right)}{\frac{9}{10}}\\ S_8=\frac{10000\left(\frac{99999999}{100000000}\right)}{\frac{9}{10}}\\ S_8=\left(\frac{11111111}{1000}\right)=11111.111\end{array}$

Conclusion:The given series is a geometric series and sum of its first eight terms is 11111.111.