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.
Find the first five terms and the sum of an infinite geometric sequence having $a_2=2$ and $a_3=1$
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:
$$4,2,1, \dfrac{1}{2}, \dfrac{1}{4}, \ldots$$