a geometric series has a sum of 1365 each term increases by a factor of 4 if there are 6 terms find the value of the first terms

Let's solve the problem step by step.

### Given:
- Sum of the series (\(S_n\)) = 1365
- Common ratio (\(r\)) = 4
- Number of terms (\(n\)) = 6

We need to find the value of the first term (\(a_1\)).

### Formula:
The sum \(S_n\) of a geometric series is given by:
\[
S_n = a_1 \times \frac{r^n - 1}{r - 1}
\]

### Substituting the Given Values:
\[
1365 = a_1 \times \frac{4^6 - 1}{4 - 1}
\]

First, calculate \(4^6\):
\[
4^6 = 4096
\]

Now, substitute and simplify:
\[
1365 = a_1 \times \frac{4096 - 1}{3}
\]
\[
1365 = a_1 \times \frac{4095}{3}
\]
\[
1365 = a_1 \times 1365
\]

### Solve for \(a_1\):
\[
a_1 = \frac{1365}{1365} = 1
\]

So, the value of the first term is **1**.

Learn how to find the value of the first term in a geometric series where the sum is 1365 and each term increases by a factor of 4. This problem involves calculating the sum of a geometric series formula and understanding exponential growth. Suitable for high school students.

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