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 + m x ) 50 is ( 3 n + 1 ) ? C 3 51 for some positive integer n . Then the value of n is

Question

Let $m$ be the smallest positive integer such that the coefficient of $x^{2}$ in the expansion of ${(1 + x)}^{2} + {(1 + x)}^{3} + \dots + {(1 + x)}^{49} + {(1 + m x)}^{50}$ is $(3 n + 1) {}^{51}C_{3}$ for some positive integer $n$ . Then the value of $n$ is [2016]

Smart Achievers · Accepted Answer

(5)

${(1 + x)}^{2} + {(1 + x)}^{3} + \dots + {(1 + x)}^{49} + {(1 + m x)}^{50}$

$= {(1 + x)}^{2} [\frac{{(1 + x)}^{48} - 1}{(1 + x) - 1}] + {(1 + m x)}^{50}$

$= \frac{1}{x} [{(1 + x)}^{50} - {(1 + x)}^{2}] + {(1 + m x)}^{50}$

$Coeff. of x^{2} in the above expansion$

$= Coeff. of x^{3} in {(1 + x)}^{50} + Coeff. of x^{2} in {(1 + m x)}^{50}$

$= {}^{50}C_{3} + {}^{50}C_{2} m^{2}$

$∴$ $(3 n + 1) {}^{51}C_{3} = {}^{50}C_{3} + {}^{50}C_{2} m^{2}$

$\Rightarrow (3 n + 1) = \frac{{}^{50}C_{3}}{{}^{51}C_{3}} + \frac{{}^{50}C_{2}}{{}^{51}C_{3}} m^{2}$

$\Rightarrow 3 n + 1 = \frac{16}{17} + \frac{1}{17} m^{2}$

$\Rightarrow n = \frac{m^{2} - 1}{51}$

$∴$ Least positive integer $m$ for which $n$ is an integer is $m = 16$ and then $n = 5$

Solution

Q.
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 [2016]

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Solution

Q. Let m be the smallest positive integer such that the coefficient of x2 in the expansion of (1+x)2+(1+x)3+⋯+(1+x)49+(1+mx)50 is (3n+1) C351 for some positive integer n. Then the value of n is [2016]

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Q.
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 [2016]