Question

The third term of a GP is 4. Find the product of its five terms.

Answer 1

In the above problem we have to find the terms of the GP.
GP is the geometric progression, which means that the numbers series or sequence has the common ratio.
$a$, $ar$, $ar^2$, $ar^3$.....................where a is the first term and r is the common ratio of the series or sequence.
Using the above equation we will perform the calculation in order to find the product of five terms.

Complete step-by-step solution:
Let's define GP in more details and then we will proceed for the calculation.
GP, geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non one number called the common ratio.
For example, 5, 10, 15, 20..................in this series the common ratio is 5. After dividing each term by the next we will get 5.
Now, we will do the calculation part.
We are given the third term of the GP.......as 4, which means ar2.
$\Rightarrow a{r^2} = 4$ ....................1
The five terms of the GP are;
$a,ar,a{r^2},a{r^3},a{r^4}$
After the product we will get;
$\Rightarrow {a^5}{r^{10}}$ ......................2
On arranging the equation 2 we will get;
$\Rightarrow {(a{r^2})^5}$......................3
From equation 1 we will substitute in equation 3
$\Rightarrow {4^5}$
$\Rightarrow 1024$ (Product of the five terms of the GP)

The product of the first 5 terms of A.P is equal to 1024.

Note: Application of GP in our daily life is; population growth in each year, Each radioactive independently disintegrates so each has a fixed term decay rate, interest rate in banks on various loan amounts, savings, fixed deposits, mutual funds which generally have compound interest growth, email chains etc.