# Question 3 Exercise 4.5

Solutions of Question 3 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

## Question 3

Find the first five terms and the sum of an infinite geometric sequence having $a_2=2$ and $a_3=1$

### Solution

We first try to have find $a_1$ and $r$.

We know that $$a_n=a_1 r^{n-1}$$

therefore

$$a_2=a_1 r=2....(i)$$

and $$a_3=a_1 r^2=1...(ii)$$

Dividing (ii) by (i), we get

\begin{align}\dfrac{a_1 r^2}{a_1 r}&=\dfrac{1}{2}\\
\Rightarrow r&=\dfrac{1}{2} \text {, }\end{align}
putting this in (i), we have

\begin{align}\dfrac{a_1}{2}&=2\\
\Rightarrow a_1&=4 \text {. }\\
a_2&=a_1 r=4 \cdot \dfrac{1}{2}=2,\\
a_3&=a_1 r^2=4 \cdot(\dfrac{1}{2})^2=1 \text {. }\\
a_4&=a_1 r^3=4(\dfrac{1}{2})^3=\dfrac{1}{2}\\
a_5&=a_1 r^4=4 \cdot(\dfrac{1}{2})^4=\dfrac{1}{4} \text {. }\end{align}
The infinite geometric sequence is:

$$4,2,1, \dfrac{1}{2}, \dfrac{1}{4}, \ldots$$

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