Question

What is the sum of a 7-term geometric series if the first term is -6, the last term is -24,576, and the common ratio is 4?

Answer 1

Answer: The required sum of the given geometric series is - 32766.

Step-by-step explanation: We are given to find the sum of a 7-term geometric series if the first term is -6, the last term is -24,576 and the common ratio is 4.

We know that,

if 'a' is the first term and 'r' is the common ratio of a geometric series, then its sum up to n terms is given by

`S_n=(a(1-r^n))/(1-r),~r<1,~~~~~\textup{or}~~~~~S_n=(a(r^n-1))/(r-1),~r>1.`

In the given geometric series,

first term, a = -6 and common ratio, r = 4.

Since r = 4 > 1, so the sum up to 7 terms is

`S_7\\\\\\=(a(r^7-1))/(r-1)\\\\\\=(-6(4^7-1))/(4-1)\\\\\\=(-6(16384-1))/(3)\\\\\\=-2* 16383\\\\=-32766.`

Thus, the required sum of the given geometric series is - 32766.

Answer 2

Hence, the sum of a 7-term geometric series is:

-32766.

Explanation:

We have to find the sum of a 7-term geometric series (i.e. n=7) if the first term(a) is -6, the last term is -24,576, and the common ratio(r) is 4.

We know that the sum of the 7-term geometric series is given as:

`S_n=a* ((r^n-1)/(r-1))`

On putting the value of a,n and r in the given formula we have:

`S_7=(-6)* ((4^7-1)/(4-1))\\\\\\S_7=-32766`

Hence, the sum of a 7-term geometric series is:

-32766.