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7.Binomial Theorem
medium
If the coefficients of ${5^{th}}$, ${6^{th}}$and ${7^{th}}$ terms in the expansion of ${(1 + x)^n}$be in $A.P.$, then $n =$
A
$7$ only
B
$14$ only
C
$7$ or $14$
D
None of these
Solution
(c) Coefficient of ${T_5} = {\,^n}{C_4},{T_6} = {\,^n}{C_5}$and ${T_7} = {\,^n}{C_6}$
According to the condition, $2\,{\,^n}{C_5} = {\,^n}{C_4} + {\,^n}{C_6}$
$ \Rightarrow \,\,2\left[ {\frac{{n!}}{{(n – 5)!5!}}} \right] = \left[ {\frac{{n!}}{{(n – 4)\,!\,4\,!}} + \frac{{n!}}{{(n – 6)\,!\,6\,!}}} \right]$
$ \Rightarrow \,\,2\left[ {\frac{1}{{(n – 5)\,5}}} \right] = \left[ {\frac{1}{{(n – 4)(n – 5)}} + \frac{1}{{6 \times 5}}} \right]$
After solving, we get $n=7$ or $14$.
Standard 11
Mathematics
