the sum of the first 6 terms of a geometric series is 15,624 and the common ratio is 5. what is the first…

Explanation:

Step1: Recall sum formula for geometric series

The sum formula for the first $n$ terms of a geometric series is $S_n=\frac{a(1 - r^n)}{1 - r}$, where $S_n$ is the sum of the first $n$ terms, $a$ is the first - term, $r$ is the common ratio, and $n$ is the number of terms.

Step2: Substitute given values into the formula

We are given that $S_6 = 15624$, $r = 5$, and $n = 6$. Substituting these values into the formula $S_n=\frac{a(1 - r^n)}{1 - r}$, we get $15624=\frac{a(1 - 5^6)}{1 - 5}$.

Step3: Simplify the right - hand side of the equation

First, calculate $5^6=15625$. Then $1-5^6=1 - 15625=-15624$, and $1 - r=1 - 5=-4$. So the equation becomes $15624=\frac{a\times(-15624)}{-4}$.

Step4: Solve for $a$

Cross - multiply to get $15624\times(-4)=a\times(-15624)$. Then $a = 4$.

Answer:

4