100% satisfaction guarantee Immediately available after payment Both online and in PDF No strings attached
logo-home
Previously searched by you
Previously searched by you
logo
Summary Sequences and Series Notes £4.89
Add to cart
Quickly navigate to
Preview
  2.  
  3. PEARSON (PEARSON)
  4.  
  5. Mathematics
  6.  
  7. Pure Mathematics

Summary

Summary Sequences and Series Notes
 29 views  1 purchase

This 4-page document clearly, but concisely, explains everything you need to know about sequences and series. No prior knowledge of the topic is assumed. I guarantee that after reading these, you will understand the topic much better than you did before.

Last document update: 3 year ago

Preview 1 out of 4  pages

Document information

  • April 18, 2021
  • April 18, 2021
  • 4
  • 2021/2022
  • Summary

Subjects

Written for

All documents for this subject (8)

Seller

avatar-seller
emmaharris1

Content preview

Sequences and Series
1 Arithmetic
1.1 Arithmetic sequences
An arithmetic sequence is one in which the difference between consecutive terms is constant throughout.
They are of the form:

𝑎, 𝑎 + 𝑥, 𝑎 + 2𝑥, … , 𝑎 + (𝑛 − 1)𝑥
…where 𝑎 is the value of the first term, and 𝑥 is the common difference.

Notice that the 𝑛th term is: 𝑎 + (𝑛 − 1)𝑥, as opposed to: 𝑎 + 𝑛𝑥. Consider the coefficients in front of 𝑥:

0, 1, 2, . . . , (𝑛 − 1)
There are 𝑛 coefficients, so there must be 𝑛 terms in the sequence above, hence the formula for the 𝑛th
term in a sequence is:

𝑢𝑛 = 𝑎 + (𝑛 − 1)𝑥
(Note: 0 arises from the first term – we can envisage it as: 𝑎 + 0𝑥 (= 𝑎).)


1.2 Arithmetic series
An arithmetic series is the sum of the first 𝑛 terms in an arithmetic sequence. It has the symbol: 𝑆𝑛 .
Using this definition, and (1.1), we have:

𝑆𝑛 = 𝑎 + (𝑎 + 𝑥) + (𝑎 + 2𝑥)+. . . +(𝑎 + (𝑛 − 2)𝑥) + (𝑎 + (𝑛 − 1)𝑥)
Summing this all together looks pretty tough. Instead, let’s ‘flip’ the terms around and put it side-by-side
with the original expression, in order to spot a pattern:

𝑆𝑛 = 𝑎 + (𝑎 + 𝑥) + (𝑎 + 2𝑥) + ⋯ + (𝑎 + (𝑛 − 2)𝑥) + (𝑎 + (𝑛 − 1)𝑥)
𝑆𝑛 = (𝑎 + (𝑛 − 1)𝑥) + (𝑎 + (𝑛 − 2)𝑥) + (𝑎 + (𝑛 − 2)𝑥)+. . . +(𝑎 + 𝑥) + 𝑎
‘Pair them up’ as shown above. You can see that each pair sums to: 2𝑎 + (𝑛 − 1)𝑥.

Since there are 𝑛 terms in the expression 𝑆𝑛 , then there are 𝑛 pairs in total as well.
If we sum the 2 expressions together, then, we get:

2𝑆𝑛 = 𝑛 × (2𝑎 + (𝑛 − 1)𝑥)
1
∴ 𝑆𝑛 = × 𝑛 × (2𝑎 + (𝑛 − 1)𝑥)
2
Remembering this formula would be horrendous, but fortunately we can see that the last term in the
sequence, call it 𝑙, was: 𝑙 = 𝑎 + (𝑛 − 1)𝑥.

∴ 2𝑎 + (𝑛 − 1)𝑥 = 𝑎 + (𝑎 + (𝑛 − 1)𝑥) = 𝑎 + 𝑙
…which is the first term + the last term of the series. And, substituting this into our expression for 𝑆𝑛 :

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

Quick and easy check-out

You can quickly pay through credit card for the summaries. There is no membership needed.

Focus on what matters

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller emmaharris1. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for £4.89. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

53008 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy revision notes and other study material for 15 years now

Start selling
£4.89  1x  sold
  • (0)
Add to cart
Added