# Numbers | Maths with Lemon

Source: https://mathswithlemon.io/aa-numbers-Series.html


## Series

"She cannot do Subtraction,” said the White Queen. “Can you do Division? Divide a loaf by a knife—what is the answer to that?” “I suppose—” Alice was beginning, but the Red Queen answered for her. “Bread-and-butter, of course." Lewis Carroll


#### Sigma Notation


##### Prior Knowledge

- Recognize all the basic functions
- You must be able to manipulate algebraic expressions. (If not, check this lesson)
- You must be able to manipulate exponential expressions. (If not, check this lesson)
- You must be able to manipulate logarithmic expressions. (If not, check this lesson)

##### Key Points

- 1. Watch the video:
- 2. Read this presentation
- 3. Watch this solved problem

#### Solved problem on Arithmetic Series


##### What you have to know:

- How to find the nth term of an Arithmetic Sequence (If not, check this lesson)

##### Key Points

- 1. Watch the video:
- 2. Read this presentation
- 2. Read this presentation

#### Sum of infinite Geometric Series


##### Prior Knowledge

- To be able to find terms and the sum of a geometric sequence.

##### Key Points

- 1. Watch the video:
- 2. Read this presentation

#### Financial applications of geometric sequences


##### What you have to know :

- You must be calculate geometric series (If not, check this lesson)
- You must be able to Rearrange Formulae. (If not, check this lesson)

##### Key Points

- 1. Watch the video:
- 2. Read this presentation

#### Extra


##### Material and references:

- Hodder Book SL(ISBN: 9781510462359) : 13A, 2B, 2C

##### Key Points

- 1. You should be able to work with arithmetic and geometric sequences given in sigma notation: \[ \sum_{r=1}^{n} u_r = u_1 + u_2 + u_3 + \cdots + u_n. \] The value of \( r \) at the bottom of the sigma (here \( r = 1 \)) shows where the counting starts, and the value of \( r \) at the top of the sigma (here \( r = n \)) shows where the counting stops.
- 2. You should be able to find the sum of infinite geometric sequences. For a geometric sequence with common ratio \( r \), \[ S_\infty = \frac{u_1}{1 - r}, \qquad |r| < 1. \]
- 3. You should be able to apply arithmetic and geometric sequences to real-world problems, often involving percentage change. A change of \( r\% \) is equivalent to multiplying by the factor \( 1 + \frac{r}{100} \). An annual inflation rate of \( i\% \) is equivalent to multiplying by \( \frac{1}{1 + \frac{i}{100}} \) each year to find the value in real terms.

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