Question

the sum of the first three terms of a GP in which the difference between the second and the first term is 6 and the difference between the fourth and the third term is 54 is​

Answer 1

Answer:-

Given:

Difference between 2nd and 1st term of a GP = 6

Difference between 4th and 3rd term = 54

We know that,

nth term of a GP = $\sf a . r^{n - 1}$

Hence,

$\sf \implies a. {r}^{2 - 1} - a = 6 \\ \\ \sf \implies \: ar - a = 6 \\ \\ \implies \sf \: a(r - 1) = 6 \\ \\ \sf \implies \: a = \dfrac{6}{r - 1} \: \: - - \: equation \: (1)$

Similarly,

$\sf \implies \: a. {r}^{4 - 1} - a. {r}^{3 - 1} = 54 \\ \\ \implies \sf \: a( {r}^{ 3} - {r}^{2} ) = 54 \: \: - - \: \: equation(1)$

Substitute the value of a from equation (1).

$\implies \sf \: \dfrac{6}{r - 1} \times {r}^{2} \times (r - 1) = 54 \\ \\ \implies \sf \: {r}^{2} = \dfrac{54}{6} \\ \\ \implies \sf \: {r}^{2} = 9 \\ \\ \sf \implies \large{ r = 3}$

Substitute the value of r in equation (1).

$\implies \sf \: a = \dfrac{6}{3 - 1} \\ \\ \sf \implies \:a = \dfrac{6}{2} \\ \\ \implies \sf \large{a = 3}$

Now,

• First term = a = 3

• Second term = ar = 3 * 3 = 9

• Third term = a * r² = 3 * (3)² = 27.

Hence,

Sum of first 3 terms = 3 + 9 + 27

→ Sum of first 3 terms = 39

Therefore, the sum of first 3 terms of the given GP is 39.