Question

The first three terms of a geometric series are `5, 15, 45`. If the nth term of this series is `10935`, what is the sum of the first n terms?

Answer 1

16400

Explanation:

This is a geometric series with r = 3. First, find what term `10935` is.

`5 * (3^(n-1)) = 10935`

`(3^(n-1)) = 2187`

`3^(n-1) = 3^7`

`n-1 = 7`

`n = 8`

Now use the formula for the sum of the first `n` terms of a geometric series:

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

`S_8 = (5(1 - 3^8)) / (1 - 3) = 16400`

Final Answer

Answer 2

The sum is `16,400`.

Explanation:

Step 1: Classify the sequence

Since `t_2 = 3t_1` and `t_3 = 3t_2`, this sequence is geometric with `r = 3`.

Step 2: Find the number of terms

There is no formula we can use to evaluate the sum without knowing the number of terms. By the formula `t_n = a(r)^(n - 1)`, we have:

`10935 = 5(3)^(n - 1)`

`2187 = 3^(n - 1)`

`3^7 = 3^(n - 1)`

`7 = n - 1`

`n = 8`

Step 3: Evaluate the sum

The formula for the sum of a geometric series is `s_n = (a(1 - r^n))/(1 - r)`.

`s_8 = (5(1 - 3^8))/(1 - 3)`

`s_8 = (-32800)/(-2)`

`s_8 = 16,400`

Practice Exercises

`1`. Find the sum:

`2 + 8 + 32 +128 + ... + 524,288`

Solution

`1.`

`699,050`

Hopefully this helps!