Question

已知数列{an}满足a1=1,an+1={2an n为奇数,an+2,n为偶数},且a1+a3+a5+.+a2k--=3

Answer 1

a(n+1)=an+2 ; n is even

= 2a(n-1) + 2

a(n+1) +2= 2(a(n-1) +2)

[a(n+1) +2]/(a(n-1) +2) =2

[a(n+1) +2] / (a1+2) = 2^(n/2)

a(n+1) +2 = 3.2^(n/2)

a(n+1) = -2+3.2^(n/2) ; n is even

n is even => n+1 is odd

an = -2+ 3.2^[(n-1)/2] ; n is odd

a1+a3+a5+.+a(2k-1)=3049

(-2+3.2^(0))+ (-2+3.2^(1)) +(-2+3.2^(2))+.+(-2+3.2^(k-1)) =3049

-2k + 3(2^k-1) =3049

k=10