作业帮用归纳法证明该数列的和shizhengquede
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/07 08:14:37
用归纳法证明该数列的和shizhengquede
n=1
LS = 1+cosx
RS = 1/2 + sin(3x/2)/ [2sin(x/2)]
=1/2 + sin(3x/2)/ [2sin(x/2)]
= 1/2 + [ sinx.cos(x/2) + cosx.sin(x/2) ]/[2sin(x/2)]
= 1/2 + (1/2) [sinx cot( x/2) + cosx ]
= 1/2 + (1/2) [2sin(x/2)cos(x/2).cot( x/2) + cosx ]
= 1/2 + (1/2) [2[cos(x/2)]^2 + cosx ]
= 1/2 + (1/2) [1+cosx + cosx ]
= 1+ cosx =LS
p(1) is true
Assume p(k) is true
1+cosx+...+cos(kx) = 1/2 + sin[(2k+1)x/2]/[2sin(x/2)]
for n=k+1
LS
=1+cosx+...+cos(kx) + cos(k+1)x
= 1/2 + sin[(2k+1)x/2]/[2sin(x/2)] + cos(k+1)x
= 1/2 + { sin[(2k+1)x/2] +2sin(x/2)cos(k+1)x } /[2sin(x/2)]
= 1/2 + { sin(k+1)xcos(x/2) - cos(k+1)x.sin(x/2) +2sin(x/2)cos(k+1)x } /[2sin(x/2)]
= 1/2 + { sin(k+1)xcos(x/2) + cos(k+1)x.sin(x/2) } /[2sin(x/2)]
= 1/2 + sin[(2k+3)x/2] /[2sin(x/2)]
=RS
By principle of MI,it is true for all +ve integer n