mathematical induction的题目

oceanheng

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谁能帮我解决这两题?

cheesit92

further maths T past year 来的哦~
我就做第六题，第八题很难type....
我let x = tita
assume the statement is true for all value is set of positive integers
summation (1,n) = (sin(2n+1)x)/(2sinx) - 1/2
summation(1,n+1) = (sin(2n+1)x)/(2sinx) - 1/2 +  cos2nx
= (sin(2n+1)x)/(2sinx) - 1/2 +  (2sinxcos2nx)/2sinx
= ((sin(2n+1)x)+(2sinxcos2nx))/(2sinx) - 1/2 ,
note that 2sinxcos2nx = sin[2nx+3x]-sin[2nx+x]   compound angle formula...
hence,
((sin(2n+1)x)+(sin[2nx+3x]-sin[2nx+x])/(2sinx) - 1/2
= sin[2nx+3x]/(2sinx) - 1/2
= (sin[2(n+1)+1]x)/(2sinx) - 1/2

second part 会吧？

cheesit92

第八题给你tips啦~ 需要用到 d^(n+1)/dx^(n+1) = (d/dx)(d^n/dx^n), 然后就是 sin x = cos (pi/2-x)