If the series {an} is arithmetic progression and the series {bn} is geometric progression, the sum of the first n items of the series {anbn} can be summed by the dislocation method.

For example:

an=2n- 1、bn=( 1/2)^(n)

Let: cn = anbn = (2n-1) × (1/2) n.

Then the sum of the first n terms of the sequence {cn} is t n, so:

TN = 1×( 1/2)+3×( 1/2)？ +5×( 1/2)? +…+(2n- 1)×( 1/2)^n

( 1/2)Tn = = = = = = = 1×( 1/2)？ +3×( 1/2)? +…+(2n-3)×( 1/2)^n+(2n- 1)×( 1/2)^(n+ 1)

When subtracting these two types, please pay attention to the in braces, so:

( 1/2)Tn = 1×( 1/2)+2×( 1/2)？ +2×( 1/2)? +…+2×( 1/2)^n-(2n- 1)×( 1/2)^(n+ 1)

What is in braces can be summarized by geometric series.

( 1/2)tn=( 1/2)+ 1-(2n+3)×( 1/2)^(n+ 1)

Get:

Tn=3-(2n+3)×( 1/2)^n