Question

The nth term of a G.P is 128 and the sum of its n terms is 255. If its common ratio is 2, then find its first term.

Answer 1

Hint: We can see that the nth term of a Geometric Progression, and its common ratio is given in the question. Also, since the sum of n terms is given, substitute the given values in the formula for the sum of n terms and find the first term of the sequence.
Formula Used:
The sum of first n terms of a G.P if its common ratio r is greater than 1, is given as,
 $ {S_n} = \dfrac{{a({r^n} - 1)}}{{r - 1}} $ where $ a $ is the first term of the given Geometric progression.

Complete step-by-step answer:
A sequence (list) of numbers is said to be a Geometric Progression if the ratio between any two consecutive numbers is same. This ratio is said to be the common ratio. We can often find the entire sequence whose first term is a and common ratio is r to be $ a,ar,a{r^2}... $
Here in this question, we are given that the nth term of the sequence is $ {T_n} = 128 $ and the sum of n terms is $ {S_n} = 255 $ . Also, the common ratio of the sequence $ r = 2 $ . And we are asked to find the first term of the sequence. We will use the formula for the sum of n terms of a G.P to get the first term.
Since, here the common ration $ r = 2 > 1 $ we will use the formula $ {S_n} = \dfrac{{a({r^n} - 1)}}{{r - 1}} $ where $ a $ is the first term of the given Geometric progression.
On substituting the values on the above formula, we get,
 $ 255 = \dfrac{{a({2^n} - 1)}}{{2 - 1}} $ .
 $ \Rightarrow 255 = a({2^n} - 1) $ - - - - - - - - - - - - - - - - - - - - - - (1)
But to proceed and find the value of a we need the value of n. We have got an additional clue that the nth term of the sequence is 128. We know that $ {T_n} = a{r^{n - 1}} $ .
 $ \Rightarrow 128 = a \times {2^{n - 1}} $ - - - - - - - - - - - - - - - - - - - (2)
On dividing equation (1) by (2) we get,
 $ \dfrac{{255}}{{128}} = \dfrac{{a({2^n} - 1)}}{{a \times {2^{n - 1}}}} $
 $ \Rightarrow \dfrac{{255}}{{128}} = 2 - \dfrac{1}{{{2^{n - 1}}}} $
 $ \Rightarrow \dfrac{1}{{{2^{n - 1}}}} = 2 - \dfrac{{255}}{{128}} $
 $ \Rightarrow \dfrac{1}{{{2^{n - 1}}}} = \dfrac{{256 - 255}}{{128}} = \dfrac{1}{{128}} $
 $ \Rightarrow {2^{n - 1}} = 128 $
Now on substituting this value in (2) we get,
 $ 128 = a \times 128 $
 $ \Rightarrow a = 1 $
Hence, the first term is 1.
So, the correct answer is “ 1”.

Note: Usually, people further solve to find the value of n. But here in this question on keep observation we can find that we can reduce the steps and energy by just using the result and substituting it to get a. Always be clear on what you want to find and that will save a lot of time and make the process easier.