Question

The nth term of GP is 128 and the sum of n terms is 255. If it's common ratio is 2. Find the first term?

Answer 1

a×(2^(n-1))=128
Solving we get
a2^n=256

a×((2^n)-1)/(2-1) = 255
Solving we get
a2^n-a = 255
Putting Value of a2^n
We get a = 1 which is the first term

Answer 2

Answer:

The first term of the Geometric progression is 1.

Solution:

Given, the nth term of GP ($t_n$) = 128.

And, Sum of n terms ($S_n$) = 255.

Common ratio (r) = 2.

Let the first term of the Progression be a.

We know,  

nth term of a GP =

$\begin{array} { l } { \text { a. } r ^ { n - 1 } = 128 } \\\\ { \text { a. } \frac { r ^ { n } } { r } = 128 } \end{array}$

$\begin{aligned} a r ^ { n } & = 128 r \\\\ a r ^ { n } & = 128 \times 2 \\\\ a r ^ { n } & = 256 \ldots . ( i ) \end{aligned}$

And,    

Sum of n terms of the GP = 255

$\begin{array} { l } { \frac { a \left( r ^ { n } - 1 \right) } { r - 1 } = 255 } \\\\ { \frac { a r ^ { n } - a } { 2 - 1 } = 255 } \end{array}$

From (i), Putting $ar^n$=256, we get,  

$\begin{array} { c } { \frac { 256 - a } { 1 } = 255 } \\\\ { 256 - a = 255 } \\\\ { a = 265 - 255 } \\\\ { a = 1 } \end{array}$

Hence, the first term of the Geometric progression is 1.