Binomial Theorem And Its Simple Applications Question 63
Question: If $ {{(1+x)}^{n}}=C_0+C_1x+C_2x^{2}+……….+C_{n}x^{n} $ , then $ \frac{C_1}{C_0}+\frac{2C_2}{C_1}+\frac{3C_3}{C_2}+….+\frac{nC_{n}}{{C_{n-1}}}= $
[BIT Ranchi 1986; RPET 1996, 97]
Options:
A) $ \frac{n(n-1)}{2} $
B) $ \frac{n(n+2)}{2} $
C) $ \frac{n(n+1)}{2} $
D) $ \frac{(n-1)(n-2)}{2} $
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Answer:
Correct Answer: C
Solution:
- $ \frac{C_1}{C_0}+2.\frac{C_2}{C_1}+3.\frac{C_3}{C_2}+…..+n.\frac{C_{n}}{{C_{n-1}}} $
$ =\frac{n}{1}+2\frac{n(n-1)/1.2}{n}+3\frac{n(n-1)(n-2)/3.2.1}{n(n-1)/1.2}+….+n.\frac{1}{n} $
$ =n+(n-1)+(n-2)….+1=\sum{n=\frac{n(n+1)}{2}} $
Trick: Put $ n=1,2,3 $ ….., then $ S_1=\frac{^{1}C_1}{^{1}C_0}=1 $ , $ S_2=\frac{^{2}C_1}{^{2}C_0}+2\frac{^{2}C_2}{^{2}C_1}=\frac{2}{1}+2.\frac{1}{2}=2+1=3 $
By option, (put n=1,2……) (a) and (b) does not hold condition, but C $ \frac{n(n+1)}{2} $ , put n =1, 2…… $ S_1=1,S_2=3 $ which is correct.
BETA
