evaluate the following

Question

20 ∑ n=o (-1)ⁿ2n 

Answer 1

n=0 to 20 for (-1)^n*2n means that there are 21 terms

 

20

∑(-1)^n*2n gives the following terms:

0

 

0-2+4-6+8-10+12-14+16-18+20-22+24-26+28-30+32-34+36-38+40

adding the 1st and last terms gives 0+40=40

adding the 2nd and next to last terms gives -2+(-38)=-40

adding the 3rd and next to the next to last term gives 4+36=40

adding the 4th term and the 4th term from the right gives -6+(-34)=-40

looking at the pattern you will have five groups of 40 and five groups of -40

adding these ten groups will give a sum of 0

there are 21 terms and the eleventh term is 20(10 terms to the left and 10 terms to the right) which is not part of any of the ten groups; therefore the sum of the sequence is 0+20=20

 

Answer 2

Essentially, the (-1)^n term only alternates the sign of the 2n term. So this summation would be

 

0-2+4-6+8-10+12-.....until you reach the 20th term. I did it a couple of times and got -20 as my solution.

 

There is a way you can use your TI-83/84 to do the summation, but it produced a different answer then doing it the long way. Check online to see how they do it.

 

Hope this helps.