Question

Find the sum of the following series. Round to the nearest hundredth if necessary.

9, plus, 18, plus, 36, plus, point, point, point, plus, 576
9+18+36+...+576
Sum of a finite geometric series:
Sum of a finite geometric series:
S, start subscript, n, end subscript, equals, start fraction, a, start subscript, 1, end subscript, minus, a, start subscript, 1, end subscript, r, to the power n , divided by, 1, minus, r, end fraction
S
n
​
=
1−r
a
1
​
−a
1
​
r
n

​

Answer 1

The sum of the series is 567.

The sum of a finite geometric series is given by the formula:

`\[ S_n = (a_1(1 - r^n))/(1 - r) \]`

where:

-
`\( S_n \)` is the sum of the series up to the n-th term,

-
`\( a_1 \)` is the first term of the series,

- r is the common ratio, and

- n is the number of terms.

In your case, the series is:

`\[ 9 + 18 + 36 + \ldots + 576 \]`

It looks like a geometric series with
`\( a_1 = 9 \) and \( r = 2 \)` (each term is obtained by multiplying the previous term by 2).

The sum
`\( S_n \)` can be calculated using the formula. Since r = 2, the formula becomes:

`\[ S_n = (9(1 - 2^n))/(1 - 2) \]`

Now, you mentioned the series goes up to 576, so we need to find n such that
`\( 9 * 2^n = 576 \).`

`\[ 2^n = (576)/(9) = 64 \]`

Taking the logarithm base 2 of both sides:

`\[ n = \log_2(64) = 6 \]`

Now, substitute n = 6 into the formula:

`\[ S_n = (9(1 - 2^6))/(1 - 2) \]`

Calculate the expression to find the sum.

`\[ S_6 = (9(1 - 64))/(-1) \]`

`\[ S_6 = (9(-63))/(-1) = (-567)/(-1) = 567 \]`

Therefore, the sum of the series is 567.