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If the coefficients of 5th, 6th , and 7t...

If the coefficients of 5th, 6th , and 7th terms in the expansion of `(1+x)^n` are in A.P., then `n=` a. 7 only b. 14 only c. 7 or 14 d. none of these

A

7 only

B

14 only

C

7 or 14

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
Coefficient of `T_(5)` is `.^(n)C_(4)`, that of `T_(6)` in `.^(n)C_(5)` and that of `T_(7)` is `.^(n)C_(6)`.
According to the condition, `2.^(n)C_(5)=.^(n)C_(4)+.^(n)C_(6)`
`:. 2[(n!)/((n-5)!5!)] = [(n!)/((n-4)!4!)+(n!)/((n-6)!6!)]`
or `2[(1)/((n-5)5)]=[(1)/((n-4)(n-5))+(1)/(6xx5)]`
After solving, we get `n = 7` or `14`.
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