Question

If the sum of the nth term of G.P. series is 255, the common ratio is 2& the last term (nth term) is 128. Find the value of n

Answer 1

Given:

• Sum of first n terms of a GP is 255
• Common ratio of GP is 2
• Last term i.e. an = 128

To find: Value of n

Solution:

We know that any term of a GP is given by arⁿ⁻¹

$\sf\implies ar^{n-1} = 128$

$\sf\implies a(2)^{n-1} = 128$

$\sf\implies a(2)^{n} = 256\qquad...Eq.(1)$

Also we know that sum of n terms of a GP is given by:

• $\boxed{\sf S_n = \dfrac{a(r^n - 1)}{r-1}}$

Here the terms used are:

• Sn = Sum of n terms
• a = First term
• r = Common ratio
• n = number of terms

By substituting the known values, we get the following results:

${\implies\sf 255= \dfrac{a(2^n - 1)}{2-1}}$

${\implies\sf 255= a(2^n - 1)}$

${\implies\sf 255= a(2)^n - a}$

Using equation(1)

${\implies\sf 255= 256 - a}$

${\implies\sf a = 256 - 255}$

$\boxed{\implies\sf a = 1}$

Put this value in eq.(1)

$\sf\implies a(2)^{n} = 256$

$\sf\implies 2^{n} = 2^8$

And this equation will give us n = 8

So the required value of n is 8