Question

What is the sum of the first five terms of the geometric sequence in whicha1 = 3 and r= 1/3?Express your answer as an improper fraction using the slash (/) key and nospaces.

Answer 1

To solve this, we can use the formula for the sum of the first n terms of a geometric sequence. The fromula is:

$S_n=a_1(\frac{1-r^n}{1-r})$

In this case, a1 = 3 and r = 1/3

Then:

$S_n=3(\frac{1-(\frac{1}{3})^n}{1-\frac{1}{3}})$

And to find the sum of the first 5 terms, we evaluate for n = 5:

$S_5=3(\frac{1-(\frac{1}{3})^5}{1-\frac{1}{3}})=3(\frac{1-\frac{1}{243}}{\frac{2}{3}})=3(\frac{\frac{242}{243}}{\frac{2}{3}})=3(\frac{3\cdot242}{2\cdot243})=3\cdot\frac{726}{486}=3\cdot\frac{121}{81}=\frac{121}{27}$

Thus, the answer is:

$S_5=\frac{121}{27}$