Question

Identify the general term of AGP.
A) $T_n=\left[a+(n-1)d\right]$
B) $T_n=r^{(n-1)}$
C) $T_n=\left[a+(n-1)d\right]r^{(n-1)}$
D) None of these

Answer 1

Hint- In AGP, i.e. Arithmetic-Geometric Progression, If we consider $a$ as the first term of AP, $d$ be the common difference of AP, and $r$ be the common ratio of GP, then AGP can be : $a,(a+d)r,(a+2d)r^2,(a+3d)r^3,....$.

Complete step-by-step answer:
In our daily life, we come across many patterns, so we should know about various patterns in our daily life. The examples of some pattern are given below:
i) 1,2,3,4,5……28,29,30
ii) $2,2^2,2^3,2^4,...$
iii) $1.2,{2.2}^2,{3.2}^2,{4.2}^3,...$
According to question,
We need to answer about the general term of AGP, so AGP can be written as:
$a,(a+d)r,(a+2d)r^2,(a+3d)r^3,....$
So, the general term of AGP is $T_n=\left[a+(n-1)d\right]r^{(n-1)}$.
Hence, option (C) is the correct answer.

Note- The general term of AGP, $T_n=\left[a+(n-1)d\right]r^{(n-1)}$ shows the behavior of AP and GP both. The $n^{th}$term of AGP is obtained by multiplying the corresponding terms of the arithmetic progression and geometric progression. For example: the numerators are in AP and denominators are in GP as shown below:
${\displaystyle \frac{1}{2}}+{\displaystyle \frac{3}{4}}+{\displaystyle \frac{5}{8}}+{\displaystyle \frac{7}{16}}+....$