The two successive terms in the expansion of
Question:

The two successive terms in the expansion of (1 + x)24 whose coefficients are in the ratio 1 : 4 are

(a) 3rd and 4th

(b) 4th and 5th

(c) 5th and 6th

(d) 6th and 7th

Solution:

For (1 + x)24 two successive terms have coefficients in ration 1 : 4

Let the two successive terms be (r + 1)th and (r + 2) terms

i. e. $T_{r+1}={ }^{24} C_{r} x^{r}$ and $T_{r+2}={ }^{24} C_{r+1} x^{r+1}$

Given that $\frac{{ }^{24} C_{r}}{{ }^{24} C_{r+1}}=\frac{1}{4}$

i. e. $\frac{(24) !}{r !(24-r) !} \times \frac{(r+1) !(24-r-1) !}{(24) !}=\frac{1}{4}$

i. e. $\frac{(r+1) r !(23-r) !}{r !(24-r)(23-r) !}=\frac{1}{4}$

i. e. $\frac{r+1}{24-r}=\frac{1}{4}$

i. e. $4 r+4=24-r$

i.e. $5 r=20$

i.e. $r=4$

i. e. $T_{r+1}=T_{4+1}=T_{5}$ and $T_{r+2}=T_{4+2}=T_{6}$ are the two terms

Hence 5th  and 6th terms.

Hence, the correct answer is option C.

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