csat-suneung 2006 Q30

csat-suneung · South-Korea · csat__math-humanities 4 marks Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms
In the expansion of the polynomial $2 ( x + a ) ^ { n }$, the coefficient of $x ^ { n - 1 }$ and the coefficient of $x ^ { n - 1 }$ in the expansion of the polynomial $( x - 1 ) ( x + a ) ^ { n }$ are equal. Find the maximum value of $a n$ for all ordered pairs $( a , n )$ satisfying this condition. (Here, $a$ is a natural number and $n$ is a natural number with $n \geqq 2$.) [4 points] 
In the expansion of the polynomial $2 ( x + a ) ^ { n }$, the coefficient of $x ^ { n - 1 }$ and the coefficient of $x ^ { n - 1 }$ in the expansion of the polynomial $( x - 1 ) ( x + a ) ^ { n }$ are equal. Find the maximum value of $a n$ for all ordered pairs $( a , n )$ satisfying this condition. (Here, $a$ is a natural number and $n$ is a natural number with $n \geqq 2$.) [4 points]
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