Question

The first term of a GP is two twice its common.find the sun in the first two terms of the progression if it's sum to infinity is 8

Answer 1

Let's denote the first term of the geometric progression (GP) as \( a \) and the common ratio as \( r \).

According to the problem, the first term \( a \) is two times its common ratio \( r \), which can be written as:

\[
a = 2r
\]

The sum to infinity of a GP is given by the formula:

\[
S_{\infty} = \frac{a}{1 - r}
\]

We are given that the sum to infinity is equal to 8:

\[
\frac{a}{1 - r} = 8
\]

Now, substituting \( a \) from the first equation into the sum to infinity equation:

\[
\frac{2r}{1 - r} = 8
\]

To solve for \( r \), we can cross-multiply:

\[
2r = 8(1 - r)
\]

Expanding the right side:

\[
2r = 8 - 8r
\]

Now, bringing all terms involving \( r \) to one side:

\[
2r + 8r = 8
\]

\[
10r = 8
\]

Dividing both sides by 10:

\[
r = \frac{8}{10} = 0.8
\]

Now that we have \( r \), we can find \( a \):

\[
a = 2r = 2 \times 0.8 = 1.6
\]

Now, we can find the sum of the first two terms of the GP. The first two terms are \( a \) and \( ar \):

1. The first term \( a = 1.6 \).
2. The second term \( ar = 1.6 \times 0.8 = 1.28 \).

Now, the sum of the first two terms \( S_2 \):

\[
S_2 = a + ar = 1.6 + 1.28 = 2.88
\]

Therefore, the sum of the first two terms of the progression is:

\[
\boxed{2.88}
\]