First slide

Binomial theorem for positive integral Index

Question

If the coefficients of three consecutive terms in the expansion of (1+x)ⁿ are in the ratio 1:7:42 then the value of n is

Easy

A

60
B

70
C

55
D

None of these

Solution

Let (r + 1)th, (r+2)th and (r + 3)th be three consecutive
terms.
Then $^{n} C_{r} :^{n} C_{r + 1} :^{n} C_{r + 2} = 1 : 7 : 42$
Now $\frac{^{n} C_{r}}{^{n} C_{r + 1}} = \frac{1}{7} \Rightarrow \frac{r + 1}{n - r} = \frac{1}{7} \Rightarrow n - 8 r = 7 . . . . . . . . (i)$
And $\frac{^{n} C_{r + 1}}{^{n} C_{r + 2}} = \frac{7}{42} \Rightarrow \frac{r + 2}{n - r - 1} = \frac{1}{6} \Rightarrow n - 7 r = 13 . . . . . . (ii)$
On solving Eqs. (i) and (ii), we get n =55

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