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Exams
Joint Entrance Examination
Mathematics
Binomial Theorem
Question 323
Question 323 of 347
Medium
MCQ
If \[n \ge 2\] is a positive integer, then the sum of the series \[{}^{n + 1}{C_2} + 2\left( {{}^2{C_2} + {}^3{C_2} + {}^4{C_2} + ... + {}^n{C_2}} \right)\] is :
A
\[{{n(2n + 1)(3n + 1)} \over 6}\]
B
\[{{n(n + 1)(2n + 1)} \over 6}\]
C
\[{{n{{(n + 1)}^2}(n + 2)} \over {12}}\]
D
\[{{n(n - 1)(2n + 1)} \over 6}\]
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Official solution
\[{}^{n + 1}{C_2} + 2\left( {{}^2{C_2} + {}^3{C_2} + {}^4{C_2} + ........ + {}^n{C_2}} \right)\] \[{}^{n + 1}{C_2} + 2\left( {{}^3{C_2} + {}^3{C_2} + {}^4{C_2} + ........ + {}^n{C_2}} \right)\] use \[\left\{ {{}^n{C_{r + 1}} + {}^n{C_r} = {}^{n + 1}{C_r}} \right\}\] \[ = {}^{n + 1}{C_2} + 2\left( {{}^4{C_3} + {}^4{C_2} + {}^5{C_3} + ........ + {}^n{C_2}} \right)\] \[ = {}^{n + 1}{C_2} + 2\left( {{}^5{C_3} + {}^5{C_2} + ........ + {}^n{C_2}} \right)\] \[\eqalign{ \& . \cr \& . \cr \& . \cr \& . \cr \& . \cr} \] \[ = {}^{n + 1}{C_2} + 2\left( {{}^n{C_3} + {}^n{C_2}} \right)\]
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n
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1
C
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1
C
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= {}^{n + 1}{C_2} + 2.{}^{n + 1}{C_3}
=
n
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1
C
2
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n
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1
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3
\[ = {{(n + 1)n} \over 2} + 2.{{(n + 1)(n)(n - 1)} \over {2.3}}\] \[ = {{n(n + 1)(2n + 1)} \over 6}\]
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