Question

Consider the geometric series 64+96+144+216+... Find the common ratio r and the sum of the first 10 terms of this series . If necessary , round to the nearest thousandth . The common ratio,

Answer 1

We are given the following geometric series

`64+96+144+216+...`

Common ratio:

The common ratio (r) of a geometric series can be found as

r = 96/64 = 1.5

you can take any two consecutive terms, the common ratio will always be the same

r = 144/96 = 1.5

r = 216/144 = 1.5

Therefore, the common ratio of the given geometric series is 1.5

Now let us find the sum of the first 10 terms of this series.

The sum of a geometric series is given by

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

Where a₁ is the first term of the series, r is the common ratio, and n is the number of terms.

For the given case

a₁ = 64

r = 1.5

n = 10

Let us substitute these values into the above formula

`\begin{gathered} S_(10)=\frac{64\cdot_{}(1-1.5^(10))}{1-1.5} \\ S_(10)=\frac{64\cdot_{}(1-57.665^{})}{-0.5} \\ S_(10)=\frac{64\cdot_{}(-56.665^{})}{-0.5} \\ S_(10)=(-3626.56)/(-0.5) \\ S_(10)=7253.12 \end{gathered}`

Therefore, the sum of the first 10 terms of this series is 7253.12