Subjective Type

The sum of some terms of G.P. is $$315$$ whose first term and the common ratio are $$5$$ and $$2$$, respectively. Find the last term and the number of terms .

Solution

Let the sum of $$n$$ terms of the G.P. be $$315.$$
It is known that, $$\displaystyle { S }_{ n }=\frac { a({ r }^{ n }-1) }{ r-1 } $$
It is given that the first term $$a$$ is 5 and common ratio $$r$$ is 2.
$$\displaystyle \therefore 315=\frac { 5({ 2 }^{ n }-1) }{ 2-1 } \\ \Rightarrow { 2 }^{ n }-1=63\\ \Rightarrow { 2 }^{ n }=64={ 2 }^{ 6 }\\ \Rightarrow { n=6 }$$
$$ \therefore$$ Last term of the G.P. $$={ 6 }^{ th }$$term $$= { ar }^{ 6-1 }=(5){ (2) }^{ 5 }=(5)(32)=160.$$
Thus, the last term of the G.P. is $$160.$$

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