Question

(1*3*5+3*5*7+5*7*9...+11*13*15)/(2*4*6+4*6*8+6*8*10...+12*14*16)

Answer 1

(2n-1)(2n+1)(2n+3) = [(2n-1)(2n+1)(2n+3)(2n+5)-(2n-3)(2n-1)(2n+1)(2n+3)]/8,
1*3*5 + 3*5*7 + ... + 9*11*13 + 11*13*15 = [1*3*5*7-(-1)*1*3*5 + 3*5*7*9-1*3*5*7 + ... +9*11*13*15-7*9*11*13 + 11*13*15*17-9*11*13*15]/8.
=[1*3*5 + 11*13*15*17]/8.
=15[1+11*13*17]/8.

(2n)(2n+2)(2n+4) = [(2n)(2n+2)(2n+4)(2n+6)-(2n-2)(2n)(2n+2)(2n+4)]/8,
2*4*6 + 4*6*8 + ... + 10*12*14 + 12*14*16 = [2*4*6*8-0 + 4*6*8*10-2*4*6*8 + ... + 10*12*14*16-8*10*12*14 + 12*14*16*18-10*12*14*16]/8
=[12*14*16*18]/8.

(1*3*5+3*5*7+5*7*9...+11*13*15)/(2*4*6+4*6*8+6*8*10...+12*14*16)
=15[1+11*13*17]/[12*14*16*18]
=15*2432/[12*14*16*18]
=5*3*19*16*8/[3*4*14*16*2*9]
=5*19/[14*9]
=95/126
=0.75396825396825396825396825396825

Answer 2

0.75396825

Answer 3

答案

667996