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Q.
If the coefficients of the three successive terms in the binomial expansion of $(1 + x)^n$ are in the ratio $1 : 7 : 42$, then the first of these terms in the expansion is :
Let $T_{2-1}, T_{r}, T_{r+1}$ be the three successive terms of $(1+x)^{n}$
$\Rightarrow { }^{n} C_{r}:{ }^{n} C_{r-1}:{ }^{n} C_{r}=1: 7: 42$
$\Rightarrow \frac{{ }^{n} C_{r-1}}{{ }^{n} C_{r-2}}=7 ; \frac{{ }^{n} C_{r}}{{ }^{n} C_{r-1}}=\frac{42}{7}=6$ $\Rightarrow \frac{n-r+2}{r-1}=7 ; \frac{n-r+1}{r}=6$
$\Rightarrow r=8, n=55$
$\Rightarrow T_{r-1}=T_{7}$
$\Rightarrow $ First of the three given terms will be $7^{ th}$