Sequences

1.1.1 Arithmetic sequence

Arithmetic sequence the next term is the previous number + the common difference (d ).

e.g. 2,4,6,8,10, … d = +2 and 2,−3,−8,−13, … d = −5

To find the common difference d , subtract two consecutive terms of an arithmetic sequence from the term that follows it, i.e. u_(n+1) − u_n.

DB 1.2 Use the following equations to calculate the n^th term or the sum of n terms.

Often the IB requires you to first find the 1^st term and/or common difference.

Finding the first term u₁ and the common difference d from other terms.

In an arithmetic sequence u₁₀ = 37 and u₂₂ = 1. Find the common difference and the first term.

Put numbers in to n^th term formula

37 = u₁ + 9d

1 = u₁ + 21d

2. Equate formulas to find d

21d −1=9d −37

12d = −36

d = −3

3. Use d to find u₁

1 − 21 · (−3) = u₁

u₁ = 64

1.1.2 Geometric sequence

Geometric sequence the next term is the previous number multiplied by the common ratio (r).

To find the common ratio, divide any term of an arithmetic sequence by the term that precedes it, i.e.

DB 1.3 & 1.8 Use the following equations to calculate the n^th term, the sum of n terms or the sum to infinity when −1 < r < 1.

Similar to questions on Arithmetic sequences, you are often required to find the 1^st term and/or common ratio first.

1.1.3 Sigma notation

Sigma notation is a way to represent the summation of any sequence — this means that it can be used for both arithmetic or geometric series. The notation shows you the formula that generates terms of a sequence and the upper and lower limits of the terms that you want to add up in this sequence.

Finding the first term u₁ and common ratio r from other terms.

Interpret the question

The sum of the first 5 terms of a geometric sequence is 3798 and the sum to infinity is 4374. Find the sum of the first 7 terms

2. Use formula for sum of n terms

3. Use formula for sum to infinity

4. Rearrange 3. for u₁

4374(1 − r ) = u₁

5. Substitute in to 2.

6. Solve for r

7. Use r to find u₁

8. Find sum of first 7 terms

1.1.4 Compound interest

Sequences can be applied to many real life situations. One of those applications is calculating the interest of a loan or a deposit. Compound interest specifically deals with interest that is applied on top of previously calculated interest. For example, if you make a deposit in a bank and reinvest the interest you will gain even more interest next time. This happens because interest is calculated not just from your initial sum, but also including your re-investments.

DB 1.4