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Binomial Coefficients

If the ratio ...

Question

If the ratio of the coefficients of rth term and (r+1)th term in the expansion of $(1 + x)^{20}$ is 1 : 2
then r $= ?$

A

4

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B

5

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C

6

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D

7

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Solution

The correct option is D $7$
Let the two successive terms be $T_{r}$ and $T_{r + 1}$ .
Therefore it is given that
$\frac{T_{r}}{T_{r + 1}} = \frac{1}{2}$
or
$2 T_{r} = T_{r + 1}$ .
$2^{20} C_{r - 1} =^{20} C_{r}$
$\frac{2}{20 - (r - 1)} = \frac{1}{r}$
$2 r = 20 - r + 1$
$3 r = 21$
$r = 7$
Hence the two consecutive terms are $T_{7}$ and $T_{8}$

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