Question

Evaluate the geometric series. Round to the nearest hundredth. Please help me!!!

Answer 1

First of all, we can factor the 2 out of the sum:

`\displaystyle \sum_(n=0)^6 2\left((4)/(3)\right)^n = 2\sum_(n=0)^6 \left((4)/(3)\right)^n`

Now, you can use the result

`\displaystyle \sum_(n=0)^k a^n = (a^(k+1)-1)/(a-1)`

So, in your case, we have

`\displaystyle \sum_(n=0)^6 \left((4)/(3)\right)^n = (\left((4)/(3)\right)^7-1)/((4)/(3)-1)=((4^7-3^7)/(3^7))/((1)/(3))=3\cdot (4^7-3^7)/(3^7) = (4^7-3^7)/(3^6)`

And finally, we have

`\displaystyle 2\sum_(n=0)^6 \left((4)/(3)\right)^n = 2\cdot(4^7-3^7)/(3^6) = (28394)/(729)\approx 38.95`