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The sum of the first 6 terms of a geometric series is 15,624 and the common ratio is 5 .
What is the first term of the series?

The sum of the first $6$ terms of a geometric series is $15$ , $624$ and the common ratio is $5$ . $\newline$ What is the first term of the series?

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Find the sum of a finite geometric series

Full solution

Q. The sum of the first

6

terms of a geometric series is

15

,

624

and the common ratio is

5

.

\newline

What is the first term of the series?

Formula Application: The sum of the first $n$ terms of a geometric series can be found using the formula $S_n = \frac{a(1 - r^n)}{(1 - r)}$ , 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. We are given that $S_6 = 15,624$ , $r = 5$ , and $n = 6$ . We need to solve for $a$ .
Plug Known Values: First, let's plug the known values into the formula for the sum of a geometric series: $\newline$ $S_6 = a(1 - 5^6) / (1 - 5)$ $\newline$ $15,624 = a(1 - 15625) / (1 - 5)$
Simplify Equation: Now, let's simplify the denominator and the numerator inside the parentheses: $\newline$ $15,624 = a(1 - 15625) / (-4)$ $\newline$ $15,624 = a(-15624) / (-4)$
Divide by Constant: Next, we simplify the right side of the equation by dividing $-15624$ by $-4$ : $15,624 = a(3906)$
Final Division: To find the value of $a$ , we divide both sides of the equation by $3906$ : $a = \frac{15,624}{3906}$
Find Value of a: Now, we perform the division to find the value of $a$ : $a = 4$

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