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:
$\dfrac{1}{2} + \dfrac{3}{4} + \dfrac{5}{8} + \dfrac{7}{{16}} + ....$