Solving a Sigma Notation Problem: Help Needed!

In summary, the conversation involves finding the value of x in a summation problem where the formula for an infinite geometric sum is used. The formula is Sn = u1(r^n - 1)/(r - 1) where Sn is the sum, u1 is the first term, r is the common ratio, and n represents the number of terms. After some confusion and calculations, it is determined that the value of x is 1/2.
  • #1
sammieh
12
0
Help!

We need to solve the following problem:

∞ r-1
find x if ∑ (3x/2) = 4
r=1


We know the answer is 1/2, but we have no idea how they got that answer. Can anyone help?

Thanks,
Sammie
 
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  • #2
it looks like things shifted when we wrote our problem. It should be the summation of (3x/2) raised to the r-1 power = 4 where r is going from 1 to infinity.

I hope this makes sense to someone!
 
  • #3
Do you know the formula for an infinite geometric sum?
 
  • #4
Is it something like

Sn = u1(r^n – 1)
---------- where Sn is really S sub n and u1 is really u sub 1?
r-1
 
  • #5
What does that formula mean? Identify u1, r, and especially n.
 
  • #6
o.k

r is the constant common ratio and u1 is the first term of the series and I think n is a term in the series??
 
  • #7
Well, first step is to find out what n is. You have to know what your formulas mean before you'll be able to use them.

You might also have another formula that is more directly useful, but you should still understand that one.
 
Last edited:
  • #8
o.k. n is the number of terms we are adding up. In our case infinity?

Right?
 
  • #9
Right--though it's not "really" infinity, you take the limit as n goes to infinity. What you have is really the formula for a finite sum, which you could tell from the n. So--if you don't have it already, what do you think the formula for an infinite sum is? In an infinite sum, r will be less than 1, so what is the limit of r^n as n goes to infinity?

Then it's just a matter of applying the formula to the series you gave and solving for x.
 
  • #10
i'm still confused. Is the formula for an infinite sum:

u1/r-1 ?

and is the limit 1 as r goes to infinity?
 
  • #11
n goes to infinity, not r--typo on my part. You're close, but not quite. As n goes to infinity, what does r^n go to? Then what does r^n - 1 go to?
 
  • #12
wouldn't r^n go to 1, but never reach 1 since it is always <1?

and if r^n goes to 1 then wouldn't r^n-1 go to 0?
 
  • #13
Pick some r < 1 and find out what it does. What is r^10? What is r^20?
 
  • #14
o.k for example if we take r=.5 then as you increase the exponent r gets closer & closer to 1. right?
 
  • #15
Well, does it?
 
  • #16
yes it does. as we increased the exponent the answer kept getting closer and closer to 1.
 
  • #17
What's 0.5^2?
 
  • #18
.5 ^ 2 = .25
 
  • #19
So can you guess what happens as you increase the exponent? Try a few more like .5^3 or .5^10.

Now you may be able to find that limit.
 
  • #20
O.K. I think I finally got it.

Sn = 4, u1=1 and r=3x/2, so

4 = 1/(1-3x/2)

solve for x and x = 1/2!

FINALLY!
 
  • #21
Yep, that's right.
 
  • #22
thanks for your help!
 

What is sigma notation?

Sigma notation is a mathematical notation used to represent the sum of a series of terms. It is denoted by the symbol Σ (sigma) and has both a lower and upper limit.

How do I solve a sigma notation problem?

To solve a sigma notation problem, you need to first identify the pattern or rule of the series. Then, plug in the values of the lower and upper limits into the expression and use basic algebra to simplify the equation.

What are the common mistakes when solving a sigma notation problem?

Some common mistakes when solving a sigma notation problem include forgetting to include the first term in the series, missing a negative sign, or making errors in simplifying the equation. It is important to carefully check each step and double-check your final answer.

Do I need to memorize all the sigma notation formulas?

No, you do not need to memorize all the sigma notation formulas. However, it is important to have a good understanding of the basic rules and patterns and be able to apply them to different problems.

Can I use a calculator to solve a sigma notation problem?

Yes, you can use a calculator to solve a sigma notation problem. However, it is still important to understand the steps and concepts involved in solving the problem to ensure accuracy and avoid making mistakes.

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