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The sum of the first $6$ terms of a geometric series is $15624$ and the common ratio is $5$ . What is the first term of the series?

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Solve system of equations: Complex word problems

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Q. The sum of the first

6

terms of a geometric series is

15624

and the common ratio is

5

. What is the first term of the series?

Define Terms: Let's denote the first term of the geometric series as $a$ , and the common ratio as $r$ . The sum of the first $n$ terms of a geometric series is given by 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, and $r$ is the common ratio. We are given that $S_6 = 15624$ and $r = 5$ .
Substitute Values: Substitute the given values into the sum formula for a geometric series to find the first term $a$ . $S_6 = \frac{a(1 - r^6)}{(1 - r)}$ $15624 = \frac{a(1 - 5^6)}{(1 - 5)}$
Calculate Exponents: Calculate the value of $5^6$ and substitute it into the equation. $\newline$ $5^6 = 15625$ $\newline$ $15624 = a(1 - 15625) / (1 - 5)$
Simplify Equation: Simplify the equation by performing the operations inside the parentheses. $\newline$ $15624 = a(1 - 15625) / (-4)$ $\newline$ $15624 = a(-15624) / (-4)$
Divide and Solve: Simplify the equation further by dividing $-15624$ by $-4$ . $15624 = a\left(\frac{-15624}{-4}\right)$ $15624 = a(3916)$
Divide and Solve: Simplify the equation further by dividing $-15624$ by $-4$ . $15624 = a(-15624) / (-4)$ $15624 = a(3916)$ Solve for $a$ by dividing both sides of the equation by $3916$ . $a = 15624 / 3916$ $a = 4$

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