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Question

limn2n2AnnBnCnn6 equals?

A
120
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B
34
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C
54
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D
215
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Solution

The correct option is B 215
An=13+23+33+...+n3=n2(n+1)24
Bn=14+24+34+...+n4=n(n+1)(2n+1)6.3.n2+3n15
Cn=15+25+35+...+n5=n2(n+1)24.2.n2+2n13
limn2n2AnnBnCnn6
=limn1n6{2n4.(n+1)24n2(n+1)(2n+1)(3n2+3n1)30n2(n+1)2(2n2+2n1)12}
=limn12(n+1n)2130(n+1n)(2n+1n)(3n2+3n1)n2112(n+1n)2(2n2+2n1)n2
=12630212=215
Hence, option 'D' is correct.

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