# Method of differences exam question [ ]

• thomas49th
In summary, the conversation discusses using the method of differences to simplify a series and find its summation from 1 to n. By breaking the sum into three separate summations and aligning the terms, cancellation occurs and the summation can be simplified.
thomas49th
Method of differences exam question [urgent]

## Homework Statement

$$\sum^{n}_{r=1}{\frac{5r+4}{r(r+1)(r+2)}$$

Now i know that this is equal to $$\frac{2}{r} + \frac{1}{r+1} -\frac{3}{r+2}$$

I need to use the method of differences to work out the summation of the series from 1 to n.

so i substitute values in

r=1:

$$2 + \frac{1}{2} - \frac{3}{3}$$

r=2:

$$1 + \frac{1}{3} - \frac{3}{4}$$

r=3:

$$\frac{2}{3} + \frac{1}{4} - \frac{3}{5}$$

r=n-1:

$$\frac{2}{n-1} + \frac{1}{n} - \frac{3}{n+1}$$

r=n:

$$\frac{2}{n} + \frac{1}{n+1} - \frac{3}{n+2}$$But I can't see any terms that have canceled out. How do i simplify this

Thanks :)
Thomas

r=n+1, (-3/(n+3))+1/(n+2)+2/(n+1). Look at the 2/(n+1) from this, the 1/(n+1) from r=n and the -3/(n+1) from r=n-1.

To clarify, by "this" in your second line, you meant the summand, not the series sum.

You have $$\sum_{r = 1}^n (\frac{2}{r} + \frac{1}{r + 1} - \frac{3}{r + 2})$$
Break this sum into three separate summations. After that, move constant multiples outside the summation. Now, align the terms of all three summations by pulling out enough terms so that all of the sums start with the same fraction. I think you'll see that you get a lot of cancellation then.

Yup got it. Thanks alot! diagonal 3!

Cheers :)

## 1. What is the method of differences?

The method of differences is a mathematical technique used to find the general term of a sequence or series by finding the differences between consecutive terms.

## 2. How is the method of differences applied to exam questions?

The method of differences can be applied to exam questions by using it to find the general term of a given sequence or series, which can then be used to solve for a specific term or sum.

## 3. What is the difference between forward and backward differences?

Forward differences involve finding the differences between consecutive terms in a sequence or series, while backward differences involve finding the differences between consecutive terms in the reverse order of the sequence or series.

## 4. Can the method of differences be applied to any sequence or series?

Yes, the method of differences can be applied to any sequence or series, as long as the differences between consecutive terms follow a specific pattern.

## 5. What are some common mistakes when using the method of differences?

Some common mistakes when using the method of differences include not starting with the correct term, not considering both forward and backward differences, and not recognizing when the differences follow a specific pattern.

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