If the coefficients of x 4 , x 5 and x 6 in the expansion of ( 1 + x ) n are in the arithmetic progression, then the maximum value of n is:

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​ Coeff. of x 4 = n C 4 ​ Coeff. of x 5 = n C 5 ​ Coeff. of x 6 = n C 6 ​ n C 4 ​ , n C 5 ​ , n C 6 ​ … . A P 2 C 5 ​ = n C 4 ​ + n C 6 ​ 2 = n C 5 ​ n C 4 ​ ​ + n C 5 ​ n C 6 ​ ​ { n C r ​ n C r ​ ​ = r n − r + 1 ​ } 2 = n − 4 5 ​ + 6 n − 5 ​ 12 ( n − 4 ) = 30 + n 2 − 9 n + 20 n 2 − 21 n + 98 = 0 ( n − 14 ) ( n − 7 ) = 0 n m a x ​ = 14 n m i n ​ = 7 ​

Mathematics - Binomial Theorem Question with Solution | TestHub

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