IB Maths AI 1.2 Notes

Arithmetic sequences

An arithmetic sequence is a sequence in which the difference between consecutive terms is constant. This fixed change is called the common difference.

For example, in the sequence $100,200,300,400,\dots$, each term increases by $100$, so the common difference is $d=100$.

A sequence is arithmetic if

$u_2-u_1=u_3-u_2=u_4-u_3=\dots$

Key terms:

• The first term is usually written as $u_1$ or $a$.
• The common difference is written as $d$.
• The $n$th term is written as $u_n$.

So an arithmetic sequence can be written in the form $u_1,u_1+d,u_1+2d,u_1+3d,\dots$

Image reference: glasses of water increasing by a fixed amount each time.

Arithmetic sequences are useful when a quantity changes by roughly the same amount each time.

• saving the same amount each month
• rows of seats increasing by a fixed number
• repayments increasing by a fixed amount
• water volume increasing by the same amount in identical containers

In practice, real-life data is often not perfectly arithmetic. In that case, we look for an approximately constant difference and use an arithmetic model as a reasonable approximation.