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

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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

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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). ----------------------------------------------- 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 ) \(\large\color{black}{ \displaystyle a_7=a_1\cdot {\rm r}^{7-1} }\) \(\large\color{black}{ \displaystyle a_7=a_1\cdot {\rm r}^{6} }\) \(\large\color{black}{ \displaystyle -24576=(-6)\cdot {\rm r}^{6} }\) \(\large\color{black}{ \displaystyle 4096={\rm r}^{6} }\) \(\large\color{black}{ \displaystyle \sqrt[6]{4096}=\rm r }\) \(\large\color{black}{ \displaystyle4=\rm r }\) -------------------------------------------------
But, they gave you the value of r, which makes it even easier.
\(\large\color{black}{ \displaystyle {\rm S}_k=\frac{A_1\left(1-{\rm r}^k\right)}{1-{\rm r}}}\) 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)

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