If is a periodic function with a fundamental period such that . If the coefficient of in the expansion can be expressed as , find .
Notation: denotes the binomial coefficient .
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Given f ( 2 x + 3 ) + f ( 2 x + 7 ) = 2 ....(i) replacing x by (x+2)
we get f ( 2 x + 7 ) + f ( 2 x + 1 1 ) = 2 ......(ii)
Equating (i) and (ii) we get f ( 2 x + 3 ) = f ( 2 x + 1 1 ) which can be simplified as
f ( 2 x + 3 ) = f ( 2 ( x + 4 ) + 3 ) thus the function is periodic with period equal to 4 therefore t = 4
therefore now we have to find out the coefficient of m − 1 2 in expansion of ( m + m 3 b ) 1 6
thus now you can calculate it