April 14, 2024

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) − un.
DB 1.2  Use the following equations to calculate the nth term or the sum of n terms.
Arithmetic sequence

Often the IB requires you to first find the 1st 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.

  1. Put numbers in to nth 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.

Geometric sequence

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

Geometric sequence 1

Similar to questions on Arithmetic sequences, you are often required to find the 1st 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.

sigma notation
Finding the first term u1 and common ratio r from other terms.
Sigma notation 1
  1. 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

sigma notation 2

3. Use formula for sum to infinity

sigma notation 3

4. Rearrange 3. for u1

4374(1 − r ) = u1

5. Substitute in to 2.

sigma notation 4

6. Solve for r

sigma notation 5

7. Use r to find u1

sigma notation 6

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

compound interest

Where:

FV  – is the future value,

PV  – is the present value,

n     – is number of years,

k     – is the number of compounding periods per year,

r%   – is the nominal annual rate of interest

Example: A deposit of 1000$ was made in a bank with annual interest of 3% that is compounded quarterly. Calculate the balance in 5 years.

                 We can use our compound interest equation. Let’s identify the known variables.

Compound interest 1