Question 1 and 2, Exercise 4.5
Solutions of Question 1 and 2 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.
Question 1
Find the sum of the geometric series. $16+16+16+\ldots$ to 11 terms.
Solution.
In this geometric series $a_1=16$ and $r=\dfrac{16}{16}=1$. Formula of geometric series is applicable if $r\neq 1$. Thus we do directly as follows:
\begin{align*}
&16+16+16+\ldots \text{ to 11 terms}\\
=&11(16) \\
=& 176
\end{align*}
Question 2
Find the sum of the geometric series. $75+15+3+...$ to 10 terms.
Solution.
Given: $75+15+3+...$ to 10 terms.
Here $a_1= 75$, $r = \frac{15}{75} = \frac{1}{5}$, $n = 10$.
The formula to find the sume of $n$ terms of geometric series $$ S_n = \frac{a_1 \left(1 - r^n\right)}{1 - r}, \quad r\neq 1.$$ Thus \begin{align*} S_{10} &= \frac{75\left(1 - (\frac{1}{5})^{10}\right)}{1 - \frac{1}{5}} \\ &= \frac{75\left(1\right)}{0.8} \\ &=93.75 \end{align*} Hence the required sum is $93.75$.
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