The common ratio of a geometric series is (1)/(4) and the sum of the first 4 terms is 170 . What is the first term of the series?

Geometric series sum formula: The sum of the first n terms of a geometric series is given by the formula: S_n = (a_1(1 - r^n)) / (1 - r) where S_n is the sum of the first n terms, a_1 is the first term, r is the common ratio, and n is the number of terms.Given values: We are given the sum of the first 4 terms (S_4) is 170, the common ratio (r) is 1/4, and we need to find the first term (a_1).Substitute values into formula: Substitute the given values into the sum formula: 170 = (a_1(1 - (1/4)^4)) / (1 - 1/4)Simplify the equation: Simplify the equation: 170 = (a_1(1 - 1/256)) / (3/4)Isolate the term involving a_1: Multiply both sides by (3/4) to isolate the term involving a_1: 170 * (3/4) = a_1(1 - 1/256)Calculate left side of the equation: Calculate the left side of the equation: 170 * (3/4) = 127.5Equation with simplified right side: Now we have: 127.5 = a_1(1 - 1/256)Divide both sides to solve for a_1: Simplify the right side of the equation: 127.5 = a_1(255/256)Calculate the value of a_1: Divide both sides by (255/256) to solve for a_1: a_1 = 127.5 / (255/256)Perform the multiplication: Calculate the value of a_1: a_1 = 127.5 * (256/255)Perform the multiplication: a_1 = 128

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Algebra 2

Find the sum of a finite geometric series

Full solution

Q. The common ratio of a geometric series is

\frac{1}{4}

and the sum of the first

4

terms is

170

\newline

What is the first term of the series?

Geometric series sum formula: The sum of the first $n$ terms of a geometric series is given by the formula: $\newline$ $S_n = \frac{a_1(1 - r^n)}{1 - r}$ $\newline$ where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.
Given values: We are given the sum of the first $4$ terms $S_4$ is $170$ , the common ratio $r$ is $\frac{1}{4}$ , and we need to find the first term $a_1$ .
Substitute values into formula: Substitute the given values into the sum formula: $\newline$ $170 = \frac{a_1(1 - (\frac{1}{4})^4)}{1 - \frac{1}{4}}$
Simplify the equation: Simplify the equation: $170 = \left( a_1(1 - \frac{1}{256}) \right) / \left(\frac{3}{4}\right)$
Isolate the term involving $a_1$ : Multiply both sides by $(\frac{3}{4})$ to isolate the term involving $a_1$ : $\newline$ $170 \times \left(\frac{3}{4}\right) = a_1\left(1 - \frac{1}{256}\right)$
Calculate left side of the equation: Calculate the left side of the equation: $$170$ \times \left(\frac{$3$}{$4$}\right) = $127$.$5$
Equation with simplified right side: Now we have: $127.5 = a_1(1 - \frac{1}{256})$
Divide both sides to solve for $a_1$: Simplify the right side of the equation: $127.5 = a_1\left(\frac{255}{256}\right)$
Calculate the value of $a_1$: Divide both sides by $(255/256)$ to solve for $a_1$:$\newline$\[a_1 = \frac{127.5}{(255/256)}\]
Perform the multiplication: Calculate the value of $a_1$: $\newline$\[a_1 = 127.5 \times \left(\frac{256}{255}\right)\]
Perform the multiplication: Calculate the value of $a_1$: $\newline$\[a_1 = 127.5 \times \left(\frac{256}{255}\right)\]Perform the multiplication: $\newline$\[a_1 = 128\]