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Index of (r+1)th Term from End When Counted from Beginning

The expansion...

Question

The expansion of $(1 + x)^{n}$ has $3$ consecutive terms with coefficients in the ratio $1 : 2 : 3$ and can be written in the form $^{n} C_{k} :$ $^{n} C_{k + 1} :$ $^{n} C_{k + 2}$ . The sum of all possible values of $(n + k)$ is?

18

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21

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28

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32

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Solution

The correct option is A $18$
$\frac{^{n} C_{k}}{^{n} C_{k + 1}} = \frac{k + 1}{n - k} = \frac{1}{2}$
$2 k + 2 = n - k$
$n = 3 k + 2$
Similarily for $\frac{^{n} C_{k + 1}}{^{n} C_{k + 2}} = \frac{2}{3}$
$2 n = 5 k + 8$
$6 k + 4 = 5 k + 8$
$k = 4$
$n = 14$
$n + k = 18$

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