Question

What is the sum of the geometric series?

I am very confused on how to do this. If anyone could help me understand (Go through the steps), that would be great!

I'll put the equation put in one second..

Answer 1

\[\sum_{n-1}^{10} 6(2)^2\]

Answer 2

Oops

\[\sum_{n-1}^{10} 6(2)^n\]

Answer 3

First of all, here's a reference that you might find useful:

http://www.purplemath.com/modules/series5.htm

There are 2 cases here: 1) find the sum of the first n terms of the series, and 2) find the sum of the infinite series (n goes to infinity). In your case you're going from n=1 to n=10.

I ask you to let n=1 right off. What is the value of the first term of this series?

Answer 4

Nicole: are you sure that you mean n-1? Could you possibly have meant n=1?

Answer 5

Yes, n=1
Sorry ha - Thought I fixed all the mistakes

Answer 6

:)

Answer 7

So, if n=1, what is the first term of the series?

Answer 8

I don't even know what \[\sum_{ }^{ }\] means so if you could first tell me that, it would be very helpful :P

Answer 9

That's the greek letter, sigma. In Math, it's a command; it tells you to add up everything that follows it.

If you write SIGMA, starting at n=1 and ending at n=5, of n, you get \[\sum_{n=1}^{5}n=1+2+3+4+5=15.\]

Answer 10

Oh, I see.
Okay so the first term would be 1 and then +2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 ??

Answer 11

The problem you've posted is \[\sum_{n=1}^{10}6(2)^n\]
which is a bit easier to do if you take that factor 6 outside:
\[6\sum_{n=1}^{10}2^n\]

Answer 12

Yes, to your question. Add up all 10 of those n values, and then multiply the resulting sum by 6. Done.

Answer 13

55 * 6 = 330 ?

Answer 14

Yes, that's great. Right on!
But this is for your EXAMPLE.
The problem you've posted is a bit different:

\[6\sum_{n=1}^{10}2^n\] Here we still let n=1, 2, 3, ...., 9, 10, producing 2^1, 2^2, ...
So y ou will obtain 10 powers of 2. You could calculate those and add them up, but boy, would that be slow!

Answer 15

Okay, so 330 is not eh finial answer. Good, I was confused ha

Answer 16

http://www.purplemath.com/modules/series5.htm provides you with a formula for the sum of a geometric series.

It looks like:
\[\sum_{i=1}^{n}r^n=\frac{ 1-r^n }{ 1-r }\]
Can y ou find that formula in this web page?

Answer 17

Yes

Answer 18

Good. Now, are you familiar with the term "common ratio"? In the posted problem, your common ratio is 2. Can you explain why that is?

Answer 19

Because that is were "r" would be

Answer 20

*where

Answer 21

I did the adding thing just to see but I didn't get the right answer :( @mathmale

Answer 22

The problem you posted is \[\sum_{i=1}^{10}6(2)^n\] and the common ratio is r. This means that every term equals the preceding term, times 2.

Answer 23

Please go back and copy onto paper the formula I gave you for the nth partial sum. Let me know when you've done that.

Answer 24

\[\sum_{i=1}^{n} r^2 = \frac{ 1 - r^2 }{ 1 -r }\]
this one?

Answer 25

Yes, except that it's r^n, not r^2. Are you clear on that? We will substitute r=2 later.

Answer 26

Oh, okay. So do I now fill in? Or no?

Answer 27

Now, replace that r with " 2 "
Replace that n with 10 (on the right side ONLY). What expression do you get ? Don't try to evaluate it.

Answer 28

\[\sum_{i-1}^{n}r^2 = \frac{ 1-2^{10} }{ 1-2 }\]

Answer 29

that's right. now, because you also had a ' 6 ' in the problem you posted, multiply both sides of your equation by 6. Note that the left side should look like\[6\sum_{i=1}^{10}2^n\]

Answer 30

where r is the common ratio and 10 is the number of terms you're adding up.

Answer 31

What do you get on the right side now?

Answer 32

Okay, um 1,023 ??

Answer 33

Then times 6 to have a final answer of 6,138 ?? :D

Answer 34

I'd just like to see:\[6\frac{ 1-2^{10} }{ 1-2 }\]

Answer 35

By reversing the order of subtraction, you could also write\[6\frac{ 2^{10}-1 }{ 2-1 }\]

Answer 36

Okay, so is 6,138 wrong then?

Answer 37

which you could either simplify or leave as is.

Answer 38

Note that \[\frac{ 6 }{ 2-1 }=6.\]

Answer 39

What does this leave you with?

Answer 40

6,138 ??

Answer 41

Looks OK to me (I haven't actually done the calculation). How did you evaluate 2^10?

Answer 42

2 to the power of 10 on my calculator

Answer 43

Great. You've done very well here. I'd suggest you review what we've discussed and be sure you can do this problem unaided.

Although i'd like to get off the 'Net now, I'd welcome further questions. Others are capable of answering them too, if you want fast results; I may be back on later this evening or sometime tomorrow morning.

Answer 44

Have a good night! Thank you!

Answer 45

My pleasure. Take care, Nicole!