itsmichelle29 one year ago 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

1. SolomonZelman

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 ) $$\large\color{black}{ \displaystyle a_7=a_1\cdot {\rm r}^{7-1} }$$ $$\large\color{black}{ \displaystyle a_7=a_1\cdot {\rm r}^{6} }$$ $$\large\color{black}{ \displaystyle -24576=(-6)\cdot {\rm r}^{6} }$$ $$\large\color{black}{ \displaystyle 4096={\rm r}^{6} }$$ $$\large\color{black}{ \displaystyle \sqrt[6]{4096}=\rm r }$$ $$\large\color{black}{ \displaystyle4=\rm r }$$ -------------------------------------------------

2. SolomonZelman

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

3. SolomonZelman

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

4. SolomonZelman

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

5. itsmichelle29

thanks

6. SolomonZelman

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

7. SolomonZelman

cu