Question

已知数列{bn}满足bn=(-1)^n n(n+1) Sn是前n项和

Answer 1

n = (-1)^n .n(n+1)

if n is even

b(n-1) +bn

= -(n-1)n + n(n+1)

= 2n

Sn = b1+b2+...+bn

= (b1+b2)+(b3+b4)+...+(b(n-1) +bn)

= 2 + 4+...+ 2n

= (n+1)n/2

if n is odd

b(n-2)+b(n-1)

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

=2(n-1)

Sn = b1+b2+...+bn

=(b1+b2)+(b3+b4)+...+(b(n-2) +b(n-1)) +bn

=[ 2 + 4+...+ 2(n-1) ] - n(n+1)

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

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

=-n( n+3)/2

S20=(20+1)20/2 =210

S21=-21( 21+3)/2 =-252