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Let m be the smallest positive integer s...

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)^4 +........+ (1+x)^49 + (1 + mx)^50` is `(3n + 1) .^51C_3` for some positive integer n. Then find the value of n.

Text Solution

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Coefficient of `x^(2)` in expansion
`= 1+.^(3)C_(2)+.^(4)C_(2)+.^(5)C_(2) + "….." + .^(49)C_(2)+.^(50)C_(2).m^(2)`
[as `.^(n)C_(r)+.^(n)C_(r-1) = .^(n+1)C_(r)`]
`= (.^(3)C_(5)+.^(3)C_(2)) + .^(4)C_(2) + .^(5)C_(2) + "…." + .^(49)C_(2) + .^(50)C_(2)m^(2)`
`= (.^(4)C_(3) + .^(4)C_(2)) + "....." + .^(50)C_(2)m^(2)`
`= .^(5)C_(3) + .^(50)C_(2)m^(2) + .^(50)C_(2)m^(2)`
`= .^(50)C_(3) + .^(50)C_(2)m^(2)+.^(50)C_(2)-.^(50)C_(2)`
`= .^(51)C_(3)+.^(50)C_(2)(m^(2)-1)`
`= (3n+1).^(51)C_(3)` (given)
`:. 3n.(51)/(3).^(50)C_(2) = .^(50)C_(2)(m^(2) - 1)`
`(m^(2)-1)/(51) = n`
Value of n is 5.
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