在数列{an}中,a(n+1)=an+2n,a1=2,则a100=?a(n+1)指第n+1项
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在数列{an}中,a(n+1)=an+2n,a1=2,则a100=?a(n+1)指第n+1项
在数列{an}中,a(n+1)=an+2n,a1=2,则a100=?
a(n+1)指第n+1项
在数列{an}中,a(n+1)=an+2n,a1=2,则a100=?a(n+1)指第n+1项
∵a(n+1)=an+2n
∴a(n+1)-an=2n (1)
an-a(n-1)=2(n-1)=2n-2 (2)
a(n-1)-a(n-2)=2(n-2)=2n-4
.
a3-a2=2×2=4=2n-2n+4
a2-a1=2=2n-2n+2 (n)
(1)+(2)+.+(n)得:
an-a1=n×(2n)+{n(0+[-2(n-1)])/2}
=(n^2)-n
a100=100^2-100+2=10000-100+2=9902
∵a1=2
∴an=(n^2)-n+2
∴a100=100^2-100+2=9902
an
=a(n-1)+2(n-1)
=a(n-2)+2(n-1)+2(n-2)
.............
=a1+2(n-1)+2(n-2)+2(n-3)+...+2*2+2*1
=2+2(1+n-1)(n-1)/2
=n^2-n+2
a100=100^2-100+2=9902
a(100)-a(99)=2*99,
a(99)-a(98)=2*98,
a(98)-a(97)=2*97,
...
...
...
a(2)-a(1)=2*1.
把这99个式子相加得：a(100)-a(1)=2*(99+98+97+…+3+2+1)=2*(99+1)*99/2=99*100=9900!
所以 a(100)=9902
可知a(n+1)-an=2n
a1=2
a2-a1=2*1
a3-a2=2*2
……
a100-a99=2*99
将上列式子左边全边起来，右边全加起来，得到
a100=2+2*(1+2+…+99)=2+9900=9902