Let m be the smallest positive integer such that the coefficient of x^2 in the expansion of 1+x^2+1+x^3+…+1+x^49+1+mx^50 is 3n+1 ^51C_3 for some…

Question

Let m be the smallest positive integer such that the coefficient of x^2 in the expansion of 1+x^2+1+x^3+…+1+x^49+1+mx^50 is 3n+1 ^51C_3 for some positive integer n. Then the value of n is

MARKS Content Team · Accepted Answer

Correct answer: 5. Coefficient of x^2 in the expansion of 1+x^2+1+x^3+…1+x^49+1+mx^50 is ^2C_2+^3C_2+…^49C_2+^50C_2m^2=3n+1^51C_3 ⇒ ^3C_3+^3C_2+…^49C_2+^50C_2m^2=3n+1^51C_3 (Use ^n C_r +^n C_(r+1) =^(n+1) C_(r+1) ) ⇒ ^50C_3+^50C_2m^2=3n+1 ^51C_3 50.49.48/6+50.49/2 m^2=3n+151.50.49/6 m^2=51n+1 must be a perfect square By trial ⇒ n=5 and m=16 (M,n∈N) ⇒ n =5

Let m be the smallest positive integer such that the coefficient of $x^{2}$ in the expansion of ${(1 + x)}^{2} + {(1 + x)}^{3} + \dots + {(1 + x)}^{49} + {(1 + m x)}^{50}$ is $(3 n + 1)^{51} C_{3}$ for some positive integer n.
Then the value of n is

Solution

More Binomial Theorem Questions

Let m be the smallest positive integer such that the coefficient of x2 in the expansion of 1+x2+1+x3+…+1+x49+1+mx50 is 3n+1 51C3 for some positive integer n.Then the value of n is

Solution

More Binomial Theorem Questions

Let m be the smallest positive integer such that the coefficient of $x^{2}$ in the expansion of ${(1 + x)}^{2} + {(1 + x)}^{3} + \dots + {(1 + x)}^{49} + {(1 + m x)}^{50}$ is $(3 n + 1)^{51} C_{3}$ for some positive integer n.
Then the value of n is