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?

−32,766
−19,662
19,662
32,766

Answer 1

You are given that:
\(\large\color{black}{ \displaystyle a_1=-6 }\)
\(\large\color{black}{ \displaystyle a_7=-24576 }\)
(and you are are also given the this series is geometric).
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Really, you don't need to know the value of the common ratio r, because
you can find it yourself based on the given \(a_1\) and \(a_7\)

\(\large\color{black}{ \displaystyle a_n=a_1\cdot {\rm r}^{n-1} }\) (i am sure you have seen that before:D )
\(\large\color{black}{ \displaystyle a_7=a_1\cdot {\rm r}^{7-1} }\)
\(\large\color{black}{ \displaystyle a_7=a_1\cdot {\rm r}^{6} }\)

\(\large\color{black}{ \displaystyle -24576=(-6)\cdot {\rm r}^{6} }\)
\(\large\color{black}{ \displaystyle 4096={\rm r}^{6} }\)
\(\large\color{black}{ \displaystyle \sqrt[6]{4096}=\rm r }\)
\(\large\color{black}{ \displaystyle4=\rm r }\)
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Answer 2

But, they gave you the value of r, which makes it even easier.

Answer 3

\(\large\color{black}{ \displaystyle {\rm S}_k=\frac{A_1\left(1-{\rm r}^k\right)}{1-{\rm r}}}\)

This is the sum of a sequence that starts from \(a_1\), and ends on \(a_k\)
(in other words for k number of terms)

Answer 4

now, just plug in your given information

\(k=7\) (since 7 is the last term)
\(A_1=-6\)
\(\rm r=4\)

and then evaluate this sum

Answer 5

thanks

Answer 6

ok, you are welcome, if you want.
If you have any questions, then please ask.

Answer 7

cu