Function+Binomial

Algebra Level 3

If f ( x ) f(x) is a periodic function with a fundamental period t t such that f ( 2 x + 3 ) + f ( 2 x + 7 ) = 2 f(2x+3) +f(2x+7) =2 . If the coefficient of m − 3 t m^{-3t} in the expansion ( m + b m 3 ) 4 t \left(m+\dfrac b{m^3}\right)^{4t} can be expressed as ( A C ) b C \dbinom AC b^C , find A + C A+C .

Notation: ( M N ) = M ! N ! ( M − N ) ! \dbinom MN = \dfrac {M!}{N!(M-N)!} denotes the binomial coefficient .


The answer is 23.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Given f ( 2 x + 3 ) + f ( 2 x + 7 ) = 2 f\left( 2x+3 \right) +f\left( 2x+7 \right) =2 ....(i) replacing x by (x+2)

we get f ( 2 x + 7 ) + f ( 2 x + 11 ) = 2 f\left( 2x+7 \right) +f\left( 2x+11 \right) =2 ......(ii)

Equating (i) and (ii) we get f ( 2 x + 3 ) = f ( 2 x + 11 ) f\left( 2x+3 \right) =f\left( 2x+11 \right) which can be simplified as

f ( 2 x + 3 ) = f ( 2 ( x + 4 ) + 3 ) f\left( 2x+3 \right) =f\left( 2(x+4)+3 \right) thus the function is periodic with period equal to 4 therefore t = 4

therefore now we have to find out the coefficient of m − 12 { m }^{ { -12 } } in expansion of ( m + b m 3 ) 16 { \left( { m }+\frac { b }{ { m }^{ 3 } } \right) }^{ 16 }

thus now you can calculate it

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Thank you man

LUCAS MACHADO - 3 years, 2 months ago

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