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Evaluate :1 + 3x + 6x ^(2) + 10 x ^(3)+…...

Evaluate `:1 + 3x + 6x ^(2) + 10 x ^(3)+…….` upto infinite term, whre `|x| lt 1.`

A

`(1)/((1-x)^(2))`

B

`(1)/(1-x)`

C

`(1)/((1+x)^(2))`

D

`(1)/((1-x)^(3))`

Text Solution

Verified by Experts

The correct Answer is:
D
Clearly, 1,3,6,10, . . . . . Is not an A.P. But, successive differences of terms form on A.P.
Let `S=1+3x+6x^(2)+10x^(3)+15x^(4)+ . . . .oo`
`rArr" "S-xS=1+2x+3x^(2)+4x^(3)+5x^(4)+ . . . . oo` . . .(i)
`rArr" "x(S-xS)=x+2x^(2)+3x^(3)+4x^(4)+ . . .` . . .(ii)
Subtracting (ii) and (i), we get
`S(1-x)^(2)=1+x+x^(2)+x^(3)+ . . ."to "oo`
`rArr" "S(1-x)^(2)=(1)/(1-x)rArrS=(1)/((1-x)^(3))`
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OBJECTIVE RD SHARMA ENGLISH-SEQUENCES AND SERIES-Chapter Test
  1. Evaluate :1 + 3x + 6x ^(2) + 10 x ^(3)+……. upto infinite term, whre |x...

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  2. Let H(n)=1+(1)/(2)+(1)/(3)+ . . . . .+(1)/(n), then the sum to n terms...

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  3. Sum of the first n terms of the series 1/2+3/4+7/8+(15)/(16)+............

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  4. If A(1),A(2) are between two numbers, then (A(1)+A(2))/(H(1)+H(2)) is ...

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  5. If the (m+1)t h ,(n+1)t h ,a n d(r+1)t h terms of an A.P., are in G.P....

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  6. Given that n arithmetic means are inserted between two sets of numbers...

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  7. If a,b, and c are in G.P then a+b,2b and b+ c are in

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  8. If in a progression a1, a2, a3, e t cdot,(ar-a(r+1)) bears a constant...

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  9. If in an AP, t1 = log10 a, t(n+1) = log10 b and t(2n+1) = log10 c then...

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  10. Find the sum of the series: 1^2-2^2+3^2-4^2+.....-2008^2+2009^2.

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  11. If 4a^(2)+9b^(2)+16c^(2)=2(3ab+6bc+4ca)," where "a,b,c are non-zero nu...

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  12. If Sn denotes the sum of n terms of an A.P. whose common difference is...

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  13. The sides of a right angled triangle are in A.P., then they are in the...

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  14. Find the sum of all the 11 terms of an AP whose middle most term is 30...

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  15. The maximum sum of the series 20+19 1/3+18 2/3+ is 310 b. 300 c. 0320 ...

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  16. If three numbers are in G.P., then the numbers obtained by adding the ...

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  17. If p ,q ,r are in A.P., show that the pth, qth and rth terms of any G....

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  18. Let a,b,c be three positive prime number. The progrrssion in which sqr...

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  19. If 1/(b-a)+1/(b-c)=1/a+1/c , then (A). a ,b ,a n dc are in H.P. (B). a...

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  20. If three numbers are in H.P., then the numbers obtained by subtracting...

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  21. The first three of four given numbers are in G.P. and their last three...

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