IN A GP if the sum of 2nd and 4th terms is 30 . The difference of 6th and 2nd term is 90 .

Find the 8 the term

I got r = √13/2
Is it correct?
Pls answer

I got the answer

ar + ar^3 = 30

ar^5 - ar = 90

I get a=3 and r=2

The GP is
3, 6, 12, 24, 48, 96, 192, 384, ...
T2+T4 = 6+24 = 30
T6-T2 = 96-6 = 90
T8 = 384

Yah you are correct

To find the 8th term of a geometric progression (GP), we need to have information about the common ratio (r). In this case, you are given the sum of the 2nd and 4th terms as 30 and the difference between the 6th and 2nd terms as 90.

Let's break down the problem step by step:

Step 1: Set up the equation for the sum of the 2nd and 4th terms:
The general formula for the sum of two terms in a GP is given by:

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

where Sn is the sum of n terms, a is the first term, and r is the common ratio.

Given that the sum of the 2nd and 4th terms is 30, we can write the equation as:

a * (r^2 - 1) / (r - 1) + a * (r^4 - 1) / (r - 1) = 30.

Simplifying this equation further, we get:

a * (r^2 - 1 + r^4 - 1) / (r - 1) = 30,
a * (r^4 + r^2 - 2) / (r - 1) = 30.

Step 2: Set up the equation for the difference between the 6th and 2nd terms:
Similar to step 1, the equation for the difference between two terms in a GP is:

an = a * r^(n-1),

where an is the nth term.

Given that the difference between the 6th and 2nd terms is 90, we can write:

a * r^5 - a * r = 90.

Step 3: Solve the system of equations:
Now, we have two equations with two unknowns (a and r). We can solve this system of equations to find the values of a and r.

Once you obtain the values of a and r, you can find the 8th term using the formula:

a * r^7.

Since you have provided the value of r as (√13/2), you can substitute this value into the formula to find the 8th term.

I hope this explanation helps you understand how to approach the problem. Double-check your calculations to ensure the correctness of your value for r.