Question 208031: Id have a question how would I find the sum of the first 20 terms od the series; 1+8+64+512....
I know that its always times 8; example: 512*8=4096 and then 4096*8...ect. but I bet there is an easier way to find the answer (I mean other than going through all numbers like that)
If possible could soemone tell me if 1.64703073*10^17 is the sum of the first 20 terms??
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Id have a question how would I find the sum of the first 20 terms od the series; 1+8+64+512....
I know that its always times 8; example: 512*8=4096 and then 4096*8...ect. but I bet there is an easier way to find the answer (I mean other than going through all numbers like that)
If possible could soemone tell me if 1.64703073*10^17 is the sum of the first 20 terms??
-----
1 = 8^0
8 = 8^1
64 = 8^2
512 = 8^3
4096 = 8^4
-----
since the 5th term is 8^4, this means that the 20th term would be 8^19 otherwise known as 8^(n-1)
n = 20, (n-1) = 19
-----
that doesn't solve for our sum though.
here you're in luck because there is a formula for finding the sum of a geometric series.
----
i found this one at:
http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_GeoSeries.xml
-----
the formula is:
----
Sn = a1 * (1 - r^n) / (1 - r)
----
Sn = Sum of the first n terms in the sequence.
a1 is the first term in the sequence.
n is the number of terms in the sequence.
r is the common ratio
-----
let's see how it works.
-----
your first 5 numbers in your geometric sequence are:
1
8
64
512
4096
-----
this sum should be: 4681
-----
let's see if it works.
Sn = a1 * (1 - r^n) / (1 - r)
a1 = 1
r = 8
n = 5
formula becomes:
Sn = 1 * (1 - 8^5) / (1 - 8)
which becomes:
Sn = 1 * (-32767) / -7 = 4681
-----
it works !!!!!
-----
using this on the full geometric series you have, we get:
Sn = a1 * (1 - r^n) / (1 - r)
-----
a1 = 1
r = 8
n = 20
-----
formula becomes:
Sn = 1 * (1 - 8^20) / (1 - 8)
= 1 * (1.152922 * 10^18) / (-7)
= 1.647031 * 10^17
-----
this is pretty close to the answer you got, if not right on.
i did it in excel to carry the answer out a few more digits.
this is what i got:
-----
1.64703072086692E+17
this number is the same as:
164703072086692000.
-----
|
|
|
