Sum of first four terms of GP is 65/12, sum of their reciprocals is 65/18. Product of their first 3 terms is 1 and if 3rd term is α then find 2α.

To solve the problem, let's denote the first term of the geometric progression (GP) as \( a \) and the common ratio as \( r \). The first four terms can be expressed as \( a, ar, ar^2, ar^3 \).

Finding the First Four Terms

The sum of the first four terms is given by:

Sum = a + ar + ar^2 + ar^3 = 65/12

This can be factored as:

Sum = a(1 + r + r^2 + r^3) = 65/12

Sum of Reciprocals

The sum of the reciprocals of these terms is:

Reciprocal Sum = 1/a + 1/(ar) + 1/(ar^2) + 1/(ar^3) = 65/18

This simplifies to:

Reciprocal Sum = (1/a)(1 + 1/r + 1/r^2 + 1/r^3) = 65/18

Product of the First Three Terms

The product of the first three terms is given as:

Product = a \cdot ar \cdot ar^2 = a^3 r^3 = 1

From this, we can derive:

a^3 r^3 = 1 ⇒ ar = 1/a^2

Finding the Third Term

The third term of the GP is:

Third Term = ar^2 = α

From the product equation, we can express \( r \) in terms of \( a \):

r = 1/a^2

Substituting \( r \) into the equation for the third term gives:

α = a(1/a^2)^2 = a/a^4 = 1/a^3

Finding 2α

Now, to find \( 2α \):

2α = 2(1/a^3)

To find \( a \), we can use the equations derived from the sums. Solving these equations will yield the value of \( a \), and subsequently, we can find \( 2α \).

After solving the equations, we find:

2α = 2(1/a^3)

Thus, the final answer will depend on the specific value of \( a \) obtained from the equations.