Sum

In a G.P., the ratio between the sum of first three terms and that of the first six terms is 125 : 152. Find its common ratio.

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#### Solution

Let a be the first term and r be the common ratio of given G.P.

Now, sum of first three terms = S_{3} = `(a(r^3-1))/(r-1)`

Now, sum of first six terms = S_{6} = `(a(r^6-1))/(r-1)`

It is given that

`((a(r^3-1))/(r-1))/((a(r^6-1))/(r-1))=125/152`

`=> (r^3-1)/(r^6-1)=125/152`

`=>(r^3-1)/((r^3)^2-(1)^2)=125/152`

`=> (r^3-1)/((r^3-1)(r^3+1))=125/152`

`=>1/(r^3+1)=125/152`

`=>r^3+1=152/125`

`=>r^3=152/125-1=(152-125)/125=27/125`

`=>r=3/5`

Hence, the common ratio is `3/5`.

Concept: Geometric Progression - Finding Sum of Their First ‘N’ Terms

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