If the coefficients of `5^(th), 6^(th)` and `7^(th)` terms in the expansion of `(1+x)^(n)` are in A.P. then `n = `

Question

If the coefficients of `5^(th), 6^(th)` and `7^(th)` terms in the expansion of `(1+x)^(n)` are in A.P. then `n = `

If the coefficients of `5^(th), 6^(th)` and `7^(th)` 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

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - 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`.