The common ratio of a geometric series is 3 and the sum of the first 8 terms is 3280 . What is the first term of the series?

Geometric Series Formula: The sum of the first n terms of a geometric series can be found using the formula S_n = 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. In this case, we know that S_8 = 3280, r = 3, and n = 8. We need to find the value of a.Plug in Known Values: Let's plug the known values into the sum formula for a geometric series: 3280 = a(1 - 3^8) / (1 - 3).Calculate Exponent: Calculate the value of 3^8 and subtract it from 1: 3^8 = 6561, 1 - 3^8 = 1 - 6561 = -6560.Calculate Denominator: Now, calculate the denominator of the fraction: 1 - 3 = -2.Substitute Values: Substitute the calculated values into the sum formula: 3280 = a(-6560) / (-2).Simplify Equation: Simplify the right side of the equation by dividing -6560 by -2: a(-6560) / (-2) = a * 3280.Solve for a: Now, solve for a by dividing both sides of the equation by 3280: a = 3280 / 3280, a = 1.

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The common ratio of a geometric series is 3 and the sum of the first 8 terms is 3280 .
What is the first term of the series?

The common ratio of a geometric series is $3$ and the sum of the first $8$ terms is $3280$ . $\newline$ What is the first term of the series?

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Algebra 2

Find the sum of a finite geometric series

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Q. The common ratio of a geometric series is

3

and the sum of the first

8

terms is

3280

\newline

What is the first term of the series?

Geometric Series Formula: 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. $\newline$ In this case, we know that $S_8 = 3280$ , $r = 3$ , and $n = 8$ . We need to find the value of $a$ .
Plug in Known Values: Let's plug the known values into the sum formula for a geometric series: $\newline$ $3280 = a(1 - 3^8) / (1 - 3)$ .
Calculate Exponent: Calculate the value of $3^8$ and subtract it from $1$ : $\newline$ $3^8 = 6561$ , $\newline$ $1 - 3^8 = 1 - 6561 = -6560$ .
Calculate Denominator: Now, calculate the denominator of the fraction: $1 - 3 = -2$ .
Substitute Values: Substitute the calculated values into the sum formula: $\newline$ $3280 = a(-6560) / (-2)$ .
Simplify Equation: Simplify the right side of the equation by dividing $-6560$ by $-2$ : $\frac{a(-6560)}{(-2)} = a \times 3280.$
Solve for a: Now, solve for a by dividing both sides of the equation by $3280$ : $\newline$ $a = \frac{3280}{3280},$ $\newline$ $a = 1.$