1.2 Arithmetic sequences and series (IB Maths: Analysis and Approaches HL): Revision notes

Section 1

What is an arithmetic sequence?

An arithmetic sequence increases or decreases by the same amount each time. That amount is the common difference $d$ : $d = u_{n+1} - u_n.$ The $n$ th term is $u_n = u_1 + (n-1)d,$ which is given in the formula booklet. For $18, 21, 24, \dots$ : $u_1 = 18$ , $d = 3$ , so $u_{25} = 18 + 24\times3 = 90$ .

To find which term first passes a value, set up an inequality such as $18 + 3(n-1) > 60$ and remember $n$ must be a positive integer.

Key termsarithmetic sequencecommon differencenth term

Common mistake

Using $u_1 + nd$ for the $n$ th term. The first term has had no differences added, so the $n$ th term has had $n-1$ .

Exam tip

Given two terms, e.g. $u_3 = 11$ and $u_8 = 31$ , subtract: $5d = 20$ , so $d = 4$ .

Section 2

How do we sum an arithmetic series?

An arithmetic series is the sum of the terms of an arithmetic sequence. The booklet gives $S_n = \frac{n}{2}\left(2u_1 + (n-1)d\right) = \frac{n}{2}(u_1 + u_n).$ Use the second form when you know the last term. For the 25-row theatre: $S_{25} = \frac{25}{2}(18 + 90) = 1350$ seats.

If you are given $S_n$ and need $n$ , you get a quadratic in $n$ . Solve it (factorise, formula or GDC) and reject any negative or non-integer solution.

Key termsarithmetic seriessum of n terms

Example

$S_n = 3195$ for $u_1 = 5$ , $d = 7$ : $\frac{n}{2}(7n + 3) = 3195 \Rightarrow 7n^2 + 3n - 6390 = 0 \Rightarrow n = 30$ .

Section 3

What does sigma notation mean?

Sigma notation is shorthand for a sum: $\sum_{r=1}^{n}(7r-2) = 5 + 12 + 19 + \dots + (7n-2).$ Substitute $r = 1, 2, 3, \dots$ to list terms. If the expression is linear in $r$ , the series is arithmetic: the coefficient of $r$ is the common difference, and the first term comes from $r = 1$ .

The number of terms is (top limit − bottom limit + 1): $\sum_{r=4}^{10}$ has 7 terms. If you use technology (a GDC's sum function) you must still be able to identify $u_1$ and $d$ .

Key termssigma notation

Common mistake

Taking the constant in $7r - 2$ as the first term. The first term is the value when $r = 1$ , i.e. 5.

Section 4

Applications: simple interest and linear growth

Simple interest is calculated only on the original amount, so the same interest is added every year and the values form an arithmetic sequence. If 4000 AED earns 3.5% simple interest, $d = 140$ and the value after $n$ years is $4000 + 140n$ .

Comparing two arithmetic models: set up an inequality such as $4000 + 140n > 5000 + 110n$ and interpret $n$ in context (whole years, first time something happens).

Key termssimple interest

Exam tip

Always answer in the language of the question: "after 34 complete years", not just "n = 34".

Section 5

Models that are nearly arithmetic

Real data are rarely perfectly arithmetic. If the differences are approximately constant (e.g. 2.6, 2.5, 2.7, 2.7), you can model the data with an arithmetic sequence using an approximate common difference, such as the mean difference $d \approx \frac{\text{last} - \text{first}}{\text{number of gaps}} = \frac{22.5 - 12.0}{4} = 2.625.$

Predictions within the data range (interpolation) are more reliable than those beyond it (extrapolation). Always comment on whether the real situation can keep growing at a constant rate.

Key termsapproximate common differenceextrapolation

Common mistake

Averaging the terms instead of the differences. The mean of 12.0 to 22.5 is not the common difference.

Must know

$u_n = u_1 + (n-1)d$ and $S_n = \frac{n}{2}(2u_1 + (n-1)d) = \frac{n}{2}(u_1 + u_n)$ (in the booklet).
In $\sum (ar + b)$ the common difference is $a$ ; find $u_1$ by putting $r$ = the lower limit.
$n$ must be a positive integer — round according to the context.
Simple interest gives an arithmetic sequence.
For nearly-arithmetic data, use a mean difference and comment on reliability, especially for extrapolation.

That's the notes covered.

Carry on to the next subtopic.