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Rina8888 [55]
3 years ago
5

Find the sum of the first four terms of the geometric sequence shown below. 4​, 4/ 3​, 4/9​, ...

Mathematics
1 answer:
user100 [1]3 years ago
3 0
It's evident that the first four terms are 4, 4/3, 4/9, and 4/27. So the fourth partial sum of the series is

S_4=4+\dfrac43+\dfrac49+\dfrac4{27}

It's as easy as adding up the fractions, but I bet this is supposed to be an exercise in taking advantage of the fact that the series is geometric and use the well-known formula for computing such a sum.

Multiply the sum by 1/3 and you have

\dfrac13S_4=\dfrac43+\dfrac49+\dfrac4{27}+\dfrac4{81}

Now subtracting this from S_4 gives

S_4-\dfrac13S_4=4-\dfrac4{81}

That is, all the matching terms will cancel. Now solving for S_4, you
have

\dfrac23S_4=4\left(1-\dfrac1{81}\right)
S_4=6\left(1-\dfrac1{81}\right)
S_4=\dfrac{480}{81}=\dfrac{160}{27}
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