# Sequences and Series Problem. Help Pleeaassee

• theintarnets
In summary, the student is stuck on a problem where they need to express a sum as a fraction of numbers in lowest terms. They tried the same method as in a similar problem, but they became confused after realizing that they needed to do partial fraction decomposition. They were then helped by their classmates who showed them how to solve the equations for A, B, and C. Once they did that, many neighboring terms in the sum canceled out and they were left with the end terms.
theintarnets
Sequences and Series Problem. Help! Pleeaassee!

## Homework Statement

I've attached the problem and my work. I'm supposed to express the sum as a fraction of numbers in lowest terms. The original statement was:
2/(1*2*3) + 2/(2*3*4) + 2/(3*4*5) + ... + 2/(100*101*102) and the answer is 2575/5151

In a similar problem, I solved it by decomposing it. So I tried the same thing with this problem, only I'm stuck because I don't know what to do to get to 2575/5151. Can someone please help me?

#### Attachments

• math problem.png
24.1 KB · Views: 474

I'm a little bit confused as to how you got your C value by plugging in n=1, you shoul have plugged in n=-2 (maybe you did and forgot to correct it because there were numbers crossed out). In any case, you have the correct partial fractions decomposition.

All you have to do now is write out the terms in the series and see what cancels out. A whole bunch of terms will cancel leaving you with a small sum that you have to calculate. I would suggest writing out the first 4 terms and the last 4 terms to see exactly which terms cancel out. You should be left with 4 numbers at the end that you sum together to get your answer!

The key is partial fraction decomposition:
$$\frac{1}{n (n + 1)(n + 2)} = \frac{A}{n} + \frac{B}{n + 1} + \frac{C}{n + 2}$$

Multiply out to get rid of denominators, move everything on one side of the equation, collect like powers of n, and equate the obtained coefficients with zero. You will get thre linear equations for A, B, and C, that you need to solve.

After that, many neighboring terms in the sum will cancel, and you will be left with only the end terms.

EDIT:
I guess you did the hard work

Now, notice you may write the general term as:
$$\left(\frac{1}{n} - \frac{1}{n + 1} \right) - \left( \frac{1}{n + 1} - \frac{1}{n + 2} \right)$$
Each term in the parentheses gives:
1/1 - 1/2 + 1/2 - 1/3 + ... + 1/n - !/(n + 1)
and
1/2 - 1/3 + 1/3 - 1/4 + ... + 1/(n + 1) - 1/(n + 2)

Do you see the cancelation?

Last edited:

Ohhhhhhhh I see. I've got it now, thank you guys soooooo much!

## 1. What is a sequence?

A sequence is a list of numbers that follow a specific pattern or rule. It can be finite or infinite, and each number in the sequence is called a term.

## 2. What is a series?

A series is the sum of the terms in a sequence. It can also be finite or infinite.

## 3. How do you determine the next term in a sequence?

The next term in a sequence can be determined by identifying the pattern or rule that the sequence follows, and then applying it to the previous term.

## 4. What is the difference between an arithmetic and geometric sequence?

An arithmetic sequence has a constant difference between each term, while a geometric sequence has a constant ratio between each term. In other words, in an arithmetic sequence, you add or subtract a fixed number to get the next term, while in a geometric sequence, you multiply or divide by a fixed number to get the next term.

## 5. How do you find the sum of an infinite series?

To find the sum of an infinite series, you can use a formula specific to the type of series (e.g. geometric, arithmetic, etc.), or you can use a technique called convergence and divergence to determine if the series will sum to a finite number or not.

• Precalculus Mathematics Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
601
• Precalculus Mathematics Homework Help
Replies
6
Views
945
• Precalculus Mathematics Homework Help
Replies
1
Views
4K
• Precalculus Mathematics Homework Help
Replies
6
Views
2K
• Precalculus Mathematics Homework Help
Replies
2
Views
1K
• Precalculus Mathematics Homework Help
Replies
17
Views
2K
• Precalculus Mathematics Homework Help
Replies
8
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
492
• Precalculus Mathematics Homework Help
Replies
4
Views
2K