The sum of the 1st nth term gp is 127.the sum of the reciprocal is 127/64,if the 1st term is 1.find n and the common ratio

r=2,n=7

Step-by-step explanation:

The formula for the sum of the nth term of a Geometric Progression is given as:

Sn = a(1-r^n)/(1-r)

Where n = Number of terms

r = Common ratio

From the above question, we are told that:

The sum of the 1st nth term gp is 127.

Hence:

127 = 1(1-r^n)/(1-r)

We are also told that:

The sum of the reciprocal is 127/64

This means the inverse, the formula is given as:

Sn = 1/a(1-r^n)/(1-r)

127/64 = 1(1-1/r^n)/(1-1/r)

= r^(1-n)(1-r^n)/(1-r)

Solving the above, we obtain:

r^(n-1) = 64

r^(n-1) = 2⁶

Hence:

r = 2

Solving for n

n - 1 = 6

n = 6 + 1

n = 7

Therefore:

r=2,n=7