The first term of a G.P., whose second term is 2 and sum to infinity is 8, will be A. 6 B. 3 C. 4 D. 1

To solve this problem, we need to recall some key concepts about geometric progressions (G.P.). In a G.P., each term after the first is found by multiplying the previous term by a constant called the common ratio, usually denoted as "r." Let's break down the information given to find the first term.

Understanding the Given Information

We know two important pieces of information:

• The second term of the G.P. is 2.
• The sum to infinity of the G.P. is 8.

Formulating the Terms

In a G.P., the first term is typically denoted as "a." Therefore, we can express the terms as follows:

• First term: a
• Second term: ar = 2

From the second term, we can express the common ratio "r" in terms of "a":

r = 2/a

Using the Sum to Infinity

The formula for the sum to infinity of a G.P. is given by:

S = a / (1 - r)

We know that this sum is equal to 8, so we can set up the equation:

8 = a / (1 - r)

8 = a / (1 - (2/a))

Solving the Equation

Let's simplify the right side:

8 = a / ((a - 2)/a) = a² / (a - 2)

Now, multiplying both sides by (a - 2) leads to:

8(a - 2) = a²

Expanding this gives us:

8a - 16 = a²

Rearranging the equation leads us to:

a² - 8a + 16 = 0

Factoring the Quadratic

This can be factored as:

(a - 4)(a - 4) = 0

Thus, we have:

a - 4 = 0

So, a = 4.

Conclusion

The first term of the G.P. is 4. Therefore, the correct answer is C. 4.