在数列{AN}中A1=2,A2=3,AN+2=[3+(-1)^N]AN/2则数列的前100项和为多少?另一题,数列0.7,0.77,0.777,……,0.777（n个7）的前10项和为多少?已知数列an中，a1=3，an+1=3an+2，则该数列的前n项和Sn=

Question

在数列{AN}中A1=2,A2=3,AN+2=[3+(-1)^N]AN/2则数列的前100项和为多少?
另一题,数列0.7,0.77,0.777,……,0.777（n个7）的前10项和为多少?
已知数列an中，a1=3，an+1=3an+2，则该数列的前n项和Sn=

Answer 1

a(n+2)=[3+(-1)^n]a(n)/2,
a(2n+2)=[3+(-1)^(2n)]a(2n)/2 = 2a(2n),
{a(2n)}是首项为a(2)=3,公比为2的等比数列.
a(2n)=3*2^(n-1),
a(2n-1+2)=a(2n+1)=[3+(-1)^(2n-1)]a(2n-1)/2=a(2n-1),
{a(2n-1)}是首项为a(1)=2,的常数数列.
a(2n-1)=2,
a(1)+a(2)+...+a(99)+a(100)=[a(2*1-1)+a(2*2-1)+...+a(2*50-1)] + [a(2*1)+a(2*2)+...+a(2*50)]
=2*50 + 3[1+2+...+2^(50-1)]
=2*50 + 3[2^(50)-1]/(2-1)
=100 + 3[2^(50)-1]
=97 + 3*2^(50)
b(n)=0.777(n个7）= 0.7[1+1/10+...+1/(10)^(n-1)]=0.7[1-(1/10)^n]/[1-(1/10)]
=(7/9)[1 - (1/10)^n],
b(1)+b(2)+...+b(10)=(7/9)*10 - (7/9)[(1/10)+(1/10)^2+...+(1/10)^(10)]
=70/9 - (7/90)[ 1 + (1/10) + ...+ (1/10)^9]
=70/9 - (7/90)[ 1 - (1/10)^(10)]/[1-(1/10)]
=70/9 - (7/81)[ 1 - (1/10)^(10)]
=70/9 - 7/81 + (7/81)(1/10)^(10)
=623/81 + (7/81)[1/10^(10)]
a(n+1)=3a(n)+2,
a(n+1)+1 = 3a(n)+3=3[a(n)+1],
{a(n)+1}是首项为a(1)+1=4,公比为3的等比数列.
a(n)+1 = 4*3^(n-1),
a(n) = 4*3^(n-1) - 1,
s(n) = a(1)+a(2)+...+a(n)=4[1+3+...+3^(n-1)] - n
=4[3^n - 1]/(3-1) - n
=2[3^n - 1] - n
=2*3^n - 2 - n