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The pth, (2p)th and (4p)th terms of an A...

The pth, `(2p)th` and `(4p)th` terms of an AP, are in GP, then find the common ratio of GP.

Text Solution

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Let `T_(n)=An+B`
` :. T_(p)=Ap+B,`
`T_(2p)=2Ap+B,T_(4p)=4Ap+B`
`:.T_(p),T_(2p)T_(4p)` are in GP.
`:.(2Ap+B)^(2)=(Ap+B)(4Ap+B)`
`implies ABp=0`
`:.B=0,A ne 0,p ne 0`
` implies` Common ratio, `r=(T_(2p))/(T_(p))=(2Ap+0)(Ap+0)=2`.
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ARIHANT MATHS-SEQUENCES AND SERIES-Exercise (Subjective Type Questions)
  1. The pth, (2p)th and (4p)th terms of an AP, are in GP, then find the co...

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  2. Find the sum of n terms of the series (a+b)+(a^(2)+ab+b^(2))+(a^(3)+a^...

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  3. The sequence of odd natural numbers is divided into groups 1,3,5,7,9,1...

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  4. Let a,b,c are respectively the sums of the first n terms, the next n t...

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  5. If the first four terms of an arithmetic sequence are a,2a,b and (a-6-...

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  6. If 1/1^2+1/2^2+1/3^2+...oo=pi^2/6 then value of 1-1/2^2+1/3^2-1/4^2+.....

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  7. If the arithmetic mean of a(1),a(2),a(3),"........"a(n) is a and b(1),...

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  8. If a(1),a(2),a(3),"........" is an arithmetic progression with common ...

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  9. If t(1)=1,t(r )-t( r-1)=2^(r-1),r ge 2, find sum(r=1)^(n)t(r ).

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  10. Prove that I(1),I(2),I(3)"..." form an AP, if (i) I(n)=int(0)^(pi)(...

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  11. Consider the sequence S=7+13+21+31+".....+"T(n) , find the value of T(...

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  12. Find value of (x+(1)/(x))^(3)+(x^(2)+(1)/(x^(2)))^(3)+"........"+(x^(n...

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  13. If a(m) be the mth term of an AP, show that a(1)^(2)-a(2)^(2)+a(3)^(2)...

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  14. If three unequel numbers are in HP and their squares are in AP, show ...

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  15. Ifa(1),a(2),a(3),"........",a(n) are in AP with a(1)=0, prove that (a(...

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  16. Balls are arranged in rows to form an equilateral triangle. The first ...

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  17. If theta(1),theta(2),theta(3),".......",theta(n) are in AP whose commo...

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  18. Show that,(1+5^(-1))(1+5^(-2))(1+5^(-4))(1+5^(-8))"....."(1+5^(-2n))=(...

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  19. Evaluate S=sum(n=0)^n(2^n)/((a^(2^n)+1) (where a>1).

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  20. Find the sum to infinite terms of the series tan^(-1)((1)/(3))+tan^(-1...

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