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If the coefficients of rth, (r+1)t h ,a ...

If the coefficients of rth, `(r+1)t h ,a n d(r+2)t h` terms in the expansion of `(1+x)^(14)` are in A.P., then `r` is/are a. 5 b. 11 c. `10` d. `9`

A

5

B

9

C

10

D

12

Text Solution

Verified by Experts

The correct Answer is:
a,b
Coefficient of rth (r+1) th and (r +2) th terms in ` ( 1 + x)^(14)` are
` ""^(14)C_(r-1), ""^(14)C_(r)`
and ` ""^(14)C_(r-1)`,respectively
Now , according to the question , ` 2 (""^(14)C_(r)) = ""^(14)C_(r-1) + ""^(14)C_(r+1)`
On dividing both sides by ` ""^(14)C_(r-1) ` , we get
`2= (""^(14)C_(r-1))/(""^(14)C_(r)) = (""^(14)C_(r+1))/(""^(14)C_(r))`
` rArr 2= (r)/(14 - r + 1) + (14 - (r +1)+1)/(r +1)`
` rArr 2 = (r)/(15-r) + (14-r)/(r+1)`
` rArr 2(15-r) (r+1)= r(r+1) + (15 - r) (14 - r)`
` rArr - 2r^(2) + 28r + 30 + 2r^(2) - 28 r + 210`
` rArr 4t^(2) - 56r + 180 = 0 rArr r^(2) - 14r + 45 = 0 `
` rArr (r-9)(r-5) = 0 `
` rArr r = 5,9`
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