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  • Give answer! If the coefficients of the three successive terms in the binomial expansion of (1+x)n are in the ratio1 : 7 : 42, then the first of these terms in the expansion is :

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 :

  • Option 1)

    6th

  • Option 2)

    7th

  • Option 3)

    8th

  • Option 4)

    9th

 

Answers (2)

As we learnt in

Properties of Binomial Theorem -

\dpi{120} \frac{^{n}c_{r-1}}{^{n}c_{r}}= \frac{r}{n-r+1} and

^{n}c_{r}+^{n}c_{r+1}= ^{n+1}c_{r+1}

-

 

 Let the three consecutive term be 

^{n}C_{r-1},\ \;^{n}C_{r},\ \;^{n}C_{r+1}

Thus \frac{^{n}C_{r+1}}{^{n}C_{r}}=\frac{n-r}{r+1}=\frac{6}{1}                        (i)

Also, \frac{^{n}C_{r}}{^{n}C_{r-1}}=\frac{7}{1}=\frac{n-r+1}{r}            (ii)

n - 7r = 6

n - 8r = - 1

Thus r = 7

First term is ^{n}C_{6}=7^{th} terms.

Correct option is 2.

 


Option 1)

6th

This is an incorrect option.

Option 2)

7th

This is the correct option.

Option 3)

8th

This is an incorrect option.

Option 4)

9th

This is an incorrect option.

Posted by

Vakul

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nCr-1 : nCr : nCr+1 = 1:7:42

r/(n-r+1) = 1/7

8r = n + 1

(r+1)/(n-r) = 7/42 = 1/6

7r = n - 6

r = 7

The correct option is B.

Posted by

shubham

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