Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

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?

View step-by-step help

View step-by-step help

Find the sum of a finite geometric series

Full solution

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 $3280$ : $\newline$ $a = \frac{3280}{3280},$ $\newline$ $a = 1.$

More problems from Find the sum of a finite geometric series

Question

Find the sum of the finite arithmetic series.

\sum_{n=1}^{10} (7n+4)

\newline

Posted 2 months ago

Question

Type the missing number in this sequence: `55,\59,\63,\text{_____},\71,\75,\79`

Posted 1 month ago

Question

Type the missing number in this sequence: `2, 4 8`, ______,`32`

Posted 2 months ago

Question

What kind of sequence is this?

2, 10, 50, 250, \ldots

Choices:Choices:

\newline

\text{[A]arithmetic}

\newline

\text{[B]geometric}

\newline

\text{[C]both}

\newline

\text{[D]neither}

Posted 2 months ago

Question

What is the missing number in this pattern?

1, 4, 9, 16, 25, 36, 49, 64, 81, \_\_\_\_

Posted 1 month ago

Question

Classify the series.

\sum_{n = 0}^{12} (n + 2)^3

\newline

\newline

\text{[A]arithmetic}

\newline

\text{[B]geometric}

\newline

\text{[C]both}

\newline

\text{[D]neither}

Posted 1 month ago

Question

Find the first three partial sums of the series.

\newline

1 + 6 + 11 + 16 + 21 + 26 + \cdots

\newline

Write your answers as integers or fractions in simplest form.

\newline

S_1 =

\newline

S_2 =

\newline

S_3 =

Posted 1 month ago

Question

Find the third partial sum of the series.

\newline

3 + 9 + 15 + 21 + 27 + 33 + \cdots

\newline

Write your answer as an integer or a fraction in simplest form.

\newline

S_3 =

Posted 1 month ago

Question

Find the first three partial sums of the series.

\newline

1 + 7 + 13 + 19 + 25 + 31 + \cdots

\newline

Write your answers as integers or fractions in simplest form.

\newline

S_1 =

\newline

S_2 =

\newline

S_3 =

Posted 2 months ago

Question

Does the infinite geometric series converge or diverge?

\newline

1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \cdots

\newline

\newline

\text{[A]converge}

\newline

\text{[B]diverge}

Posted 1 month ago