If the coefficients of three consecutive terms in the expansion of ${(1 + x)}^{n}$ are in the ratio of 1:7:42 then n is divisible by

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Solution

The correct options are
B 5
D 11
Let the consecutive terms be
$T_{r + 1}, T_{r + 2}, T_{r + 3}$
Hence
$7^{n} C_{r} =^{n} C_{r + 1}$
Hence
$\frac{7}{n - r} = \frac{1}{r + 1}$
$7 r + 7 = n - r$
$n - 8 r = 7$ ...(i)
Also
$42^{n} C_{r + 1} = 7^{n} C_{r + 2}$
Hence
$\frac{6}{n - (r + 1)} = \frac{1}{r + 2}$
Or
$6 r + 12 = n - r - 1$
$n - 7 r = 13... (i i)$
Solving i and ii, we get
$r = 6$ and $n = 55$ .

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