SOLUTION: 1 , 8 , 18 ... in the sequence above , the first term is 1 and the sum of the first n terms is equal to 3 to the power n for all n > 1 what is the fourth term in the sequence ?

Algebra ->  Sequences-and-series -> SOLUTION: 1 , 8 , 18 ... in the sequence above , the first term is 1 and the sum of the first n terms is equal to 3 to the power n for all n > 1 what is the fourth term in the sequence ?      Log On


   

Question 998671: 1 , 8 , 18 ...
in the sequence above , the first term is 1 and the sum of the first n terms is equal to 3 to the power n for all n > 1
what is the fourth term in the sequence ?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The sum of the first 2 terms, a%5B1%5D=1 and a%5B2%5D=8 , is
a%5B1%5D%2Ba%5B2%5D=1%2B8=9=3%5E2 .
The sum of the first 3 terms, going all the way to a%5B3%5D=18 , is
a%5B1%5D%2Ba%5B2%5D%2Ba%5B3%5D=9%2Ba%5B3%5D=9%2B18=27=9%5E3 .
The sum of the first 4 terms, going all the way to a%5B4%5D= the fourth term (to be found), is
a%5B1%5D%2Ba%5B2%5D%2Ba%5B3%5D%2Ba%5B4%5D=27%2Ba%5B4%5D must be 3%5E4=81 .
So 27%2Ba%5B4%5D=81-->a%5B4%5D=81-27-->a%5B4%5D=highlight%2854%29 .

EXTRA:
If a%5B1%5D%2Ba%5B2%5D%2B%22...%22%2Ba%5Bn-1%5D%2Ba%5Bn%5D=3%5En ,
then a%5B1%5D%2Ba%5B2%5D%2B%22...%22%2Ba%5Bn-1%5D=3%5E%28n-1%29 , and

a%5Bn%5D=3%5En-3%5E%28n-1%29
a%5Bn%5D=3%2A3%5E%28n-1%29-3%5E%28n-1%29
a%5Bn%5D=%283-1%29%2A3%5E%28n-1%29
a%5Bn%5D=2%2A3%5E%28n-1%29
All of that for n%3E1 , of course, as the problem specified.
For n=1, the sum of the first n=1 terms is a%5B1%5D=1 ,
which is not equal to 3%5E1=3 .
That formula a%5Bn%5D=2%2A3%5E%28n-1%29 is how I mentally calculated a%5B4%5D=2%2A3%5E3=2%2A27=54 ,
because 81-54 is the type of calculation that I have hated to do since the first grade.