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Show that,(1+5^(-1))(1+5^(-2))(1+5^(-4))...

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

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To show that \[ (1 + 5^{-1})(1 + 5^{-2})(1 + 5^{-4})(1 + 5^{-8}) \cdots (1 + 5^{-2n}) \leq \frac{5}{4} (1 - 5^{-(2(n+1))}), \] we will simplify the left-hand side step by step. ...
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ARIHANT MATHS ENGLISH-SEQUENCES AND SERIES-Exercise (Subjective Type Questions)
  1. If the first four terms of an arithmetic sequence are a,2a,b and (a-6-...

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

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

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

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

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

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  9. If the sequence a1, a2, a3,....... an ,dot forms an A.P., then prove ...

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

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

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

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  13. If theta(1),theta(2),theta(2), theta(3),"…………" Theta(n) are in AP w...

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

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

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  16. Find the sum of the series :tan^- 1(1/3)+tan^- 1(2/9)+....+tan^- 1((2^...

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  17. Find the sum to n terms, whose nth term is tan[alpha+(n-1)beta]tan (al...

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  18. If sum(r=1)^n tr=n/8(n+1)(n+2)(n+3), then find sum(r=1)^n1/(tr)dot

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  19. If s1,s2,s3 denote the sum of n terms of 3 arithmetic series whose fir...

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  20. Three friends whose ages form a G.P. divide a certain sum of money in ...

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