the sum of the first 8 terms of a geometric progression is 6560 and the common ratio is 3. the first term is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the first term of the geometric progression (GP) given that the sum of the first 8 terms is 6560 and the common ratio is 3, we can use the formula for the sum of the first n terms of a geometric progression:

\[ S_n = A \frac{(r^n - 1)}{(r - 1)} \]

Where:
- \( S_n \) is the sum of the first n terms,
- \( A \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the number of terms.

Given:
- \( S_8 = 6560 \)
- \( r = 3 \)
- \( n = 8 \)

We can substitute these values into the formula:

\[ 6560 = A \frac{(3^8 - 1)}{(3 - 1)} \]

Now, we need to calculate \( 3^8 \):

\[ 3^8 = 6561 \]

So, we can rewrite the equation:

\[ 6560 = A \frac{(6561 - 1)}{2} \]

This simplifies to:

\[ 6560 = A \frac{6560}{2} \]

Now, simplifying further:

\[ 6560 = A \cdot 3280 \]

To find \( A \), we divide both sides by 3280:

\[ A = \frac{6560}{3280} \]

Calculating this gives:

\[ A = 2 \]

Thus, the first term of the geometric progression is:

\[ \boxed{2} \]