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# 4.1. Sequences & Series: Sequences

### Learning Objectives

You should be able to

• Understand notation and terminology regarding sequences
• Be able to compute terms of a sequence from a formula for the nth term or a recursive/iterative definition
• Identify when a sequence is an arithmetic or geometric progression
• Write down a formula for the nth term of an arithmetic or geometric progression

### Get Started: What to do next

If you can pass the diagnostic test, then we believe you are ready to move on from this topic.

Our recommendation is to:
1. Test your ability by trying the diagnostic quiz. You can reattempt it as many times as you want and can leave it part-way through.
2. Identify which topics need more work. Make a note of any areas that you are unable to complete in the diagnostic quiz, or areas that you don't feel comfortable with.
3. Watch the tutorials for these topics, you can find these below. Then try some practice questions from the mini quiz for that specific topic. If you have questions, please ask on the forum.
4. Re-try the diagnostic quiz. And repeat the above steps as necessary until you can pass the diagnostic test. A pass grade is 80%.

Optionally, also see if you can apply these skills to an economic application or look at the additional resources and advanced quizzes at the bottom of the page,

### Tutorials: How to guides

#### 4.1.1. Notation, terminology & $$n^{th}$$ terms

Learn about terminology and notation to do with sequences and see how to evaluate terms of sequences from a formal for their nth term or a recursion/iterative definition.

#### 4.1.2. Arithmetic progression

$$a_n = a_1+(n-1)d$$, $$a_{n+1}=a_n+d$$, common difference $$d$$

#### 4.1.3. Geometric progression

$$a_n = ar^{n-1}$$, $$a_{n+1} = ra_n$$, common ratio $$r$$

CORRECTION: at 23:48 it should be $$a_{100}= -\frac{1}{10} \times 5^{99} = \frac{-5^{99}}{10} = -\frac{5^{98}}{2}$$

### Economic Application: GDP growth

One remarkable empirical regularity about the long-term GDP growth of developed economies is that GDP per capita grows at an approximately constant rate that does not diminish over time (Kaldor, 1957, The Economic Journal). The following application shows how geometric progressions can be a useful approximation in describing the long-term evolution of GDP given constant growth rates and how this approximation can be useful in answering several interesting related questions.

#### Economic Application Exercise

The following quiz allows you to test your understanding of the constant growth formula in the context of answering seveal different types of questions related to growth.

### Further practice & resources

You can find practice quizzes on each topic in the links above the tutorial videos, or you can take the diagnostic quiz as many times as you like to If you want to test yourself further, then try the advanced quiz linked below.

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