Question

If 1st term of gp is 1 the sum of 3rd and 5th term is 90 find common ratio and gp

Answer 1

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Toolbox:nthnth term of a G.P. =tn=a.rn−1=tn=a.rn−1

Given: 1st1st term =1 and sum of 3rd3rd and 5th5th=90=90

⇒a=1andt3+t5=90⇒a=1andt3+t5=90

We know that nthnth term of a G.P. istn=a.rn−1tn=a.rn−1

⇒t3=a.r2−1=1×r2=r2⇒t3=a.r2−1=1×r2=r2 and

t5=a.r5−1=1×r4=r4t5=a.r5−1=1×r4=r4.

∴r2+r4=90∴r2+r4=90

Let r2=xr2=x

then x+x2=90x+x2=90

⇒x2+x−90=0⇒x2+x−90=0

⇒x2+10x−9x−90=0⇒x2+10x−9x−90=0

⇒x(x+10)−9(x+10)=0⇒x(x+10)−9(x+10)=0

⇒(x−9)(x+10)=0⇒(x−9)(x+10)=0

⇒x=−10orx=9⇒x=−10orx=9

But since r2r2 cannot be negative, x=r2=9x=r2=9

⇒r=±3⇒r=±3