用归纳法证明该数列的和shizhengquede
来源:学生作业帮 编辑:作业帮 分类:数学作业 时间:2024/05/26 05:36:16
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
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
用数学归纳法证明数列成立
高二数列--用数学归纳法证明
请用放缩法证明该题,请用放缩法证明该题的第二问(请不要用数学归纳法和综合法),
数列{an}满足a1=1,设该数列的前n项和为Sn,且Sn,Sn+1,2a1成等差数列.用数学归纳法证明:Sn=(2n-
用数学归纳法证明该等式
用数学归纳法证明考察数列1,2,3,4,5,10,20,40,.该数列开头是等差数列,第五项以后是等比数列.证明:任意一
能否用归纳法证明 斐波那契数列的通项公式?
谁会用数学归纳法证明斐波那契数列的通项公式
数学归纳法怎么证明数列的单调性
用数学归纳法证明斐波那契数列公式
数列第一项二分之一,第n+1项是第n项与1的和得倒数,用数学归纳法证明偶数项的单调性
用数学归纳法证明数列的通项公式时,假设第k项满足该通项公式,那么其前k项是否也满足该通项公式?