Question

Find three numbers in GP, whose product is 8 and their sum is $\dfrac{{26}}{3}$

Answer 1

Here in this question, we have to find the sum of finite geometric series. The geometric series is defined as the series with a constant ratio between the two successive terms. Then by considering the geometric series we have found the sum of the series.

Complete step by step solution:
In mathematics we have three types of series namely, arithmetic series, geometric series and harmonic series.
The geometric series is defined as the series with a constant ratio between the two successive terms. The finite geometric series is generally represented as $a,ar,a{r^2},...,a{r^n}$, where a is first term and r is a common ratio.
The first three terms of GP is generally given as $\dfrac{a}{r}$, a and $ar$
The product of three terms is 8. So we have
$\dfrac{a}{r} \times a \times ar = 8$
On multiplying we have
$\Rightarrow {a^3} = 8$
On applying the cubic root to the above equation we have
$\Rightarrow a = 2$
As we know that the sum of three terms is $\dfrac{{26}}{3}$
So we have
$\dfrac{a}{r} + a + ar = \dfrac{{26}}{3}$
On substituting the value of a, the above equation is written as
$\Rightarrow \dfrac{2}{r} + 2 + 2r = \dfrac{{26}}{3}$
On simplifying we have

$$\Rightarrow 2 + 2r + 2{r^2} = \dfrac{{26}}{3}r \\\ \Rightarrow 6 + 6r + 6{r^2} = 26r \\\$$

Take 26r to LHS of the equation

$$\Rightarrow 6{r^2} + 6r - 26r + 6 = 0 \\\ \Rightarrow 6{r^2} - 20r + 6 = 0 \\\$$

Divide the above equation by 3
$\Rightarrow 3{r^2} - 10r + 3 = 0$
The equation can be written as

$$\Rightarrow 3{r^2} - 9r - r + 3 = 0 \\\ \Rightarrow 3r(r - 3) - 1(r - 3) = 0 \\\ \Rightarrow (r - 3)(3r - 1) = 0 \\\$$

$\Rightarrow r = 3$and $r = \dfrac{1}{3}$
When a = 2 and r = 3, the geometric sequence is $\dfrac{2}{3},2,6$
When a = 2 and r = $\dfrac{1}{3}$, the geometric sequence is $6,2,\dfrac{2}{3}$

Note: Three different forms of series are arithmetic series, geometric series and harmonic series. For the arithmetic series is the series with common differences. The geometric series is the series with a common ratio. The sum is known as the total value of the given series.