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The sum of the first 9 terms of the seri...

The sum of the first 9 terms of the series `1^3/1 + (1^3 + 2^3)/(1+3) + (1^3 + 2^3 +3^3)/(1 + 3 +5)` ..... is :

A

71

B

96

C

142

D

192

Text Solution

Verified by Experts

The correct Answer is:
B
Write the nth term of the given series and simplify it to get its lowest form. Then, apply `S_(n) = sum T_(n)`
Given series is `(1^(3))/(1) + (1^(3) + 2^(3))/(1 + 3) + (1^(3) + 2^(3) + 3^(3))/(1 + 3 + 5) +`....
Let `T_(n)` be the nth term of the given series.
`:. T_(n) = (1^(3) + 2^(3) + 3^(3) + ...+ n^(3))/(1 + 3 + 5 + ..+ " upton n terms")`
`= ({(n (n+1))/(2)}^(2))/(n^(2)) = ((n +1)^(2))/(4)`
`S_(9) = underset(n =1)overset(9)sum ((n+1)^(2))/(4) = (1)/(4) (2^(2) + 3^(2) + ...+ 10^(2)) + 1^(2) - 1^(2)]`
`= (1)/(4) [(10(10 + 1) (20 +1))/(5) -1] = (384)/(4) = 96`
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