Analyze arithmetic sequences and series in context

Alberta Grade 11 Mathematics

Relations and Functions

A constant difference creates a discrete pattern

An arithmetic sequence is a list of terms that changes by the same amount each time. Mathematics 20-1 also asks for the sum of a finite arithmetic series. A sequence plot contains separate points at whole-number term positions; joining the dots as a continuous curve would suggest terms that are not actually in the list. The figure below calculates each point from the first term and common difference.

What you will learn

Identify a common difference.
Find a term by position.
Find the sum of a finite arithmetic series.

From repeated addition to a term rule

The list 4, 7, 10, 13, 16 grows by 3 each time. Its first term is 4 and its common difference is 3. By the time you reach term n, you have added the difference n − 1 times.

Only whole-number term positions such as 1, 2, and 3 belong to this discrete sequence. In a context, check whether the first term corresponds to the starting state or after one period; an off-by-one error changes every later value.

a_n=a_1+(n-1)d

Pair terms to find the sum

For a finite arithmetic series, the first and last terms average to

(a_1+a_n)/2

. Multiplying that average by the number of terms gives the total.

If the last term is not provided, find it with the term rule first. The same sum can be checked by direct addition for a short list, which is useful for catching a mistaken number of terms.

S_n=

\frac{n(a_1+a_n)}{2}

Worked example

Find a term and a total

An arithmetic sequence starts 4, 7, 10, 13, 16, ... . Find the eighth term and the sum of the first eight terms.

First eight arithmetic-sequence terms

Separate dots show term positions 1 through 8; each term is three greater than the previous one.

Find the common difference
Subtract consecutive terms: 7 − 4 = 3 and 10 − 7 = 3.
$d=3$
Calculate term eight
There are seven steps of size 3 from term one to term eight.
$a_8=4+(8-1)3=25$
Calculate the sum
Average the first and eighth terms and multiply by eight.
S_8= $\frac{8(4+25)}{2}$ =116

Answer: The eighth term is 25 and the first eight terms sum to 116.

Check: Direct addition of 4 + 7 + 10 + 13 + 16 + 19 + 22 + 25 also gives 116.

Common mistakes and how to avoid them

Using n differences to reach term n.

Correction: From term one to term n there are n − 1 steps.

Using only the last term for the average.

Correction: Average the first and last terms, then multiply by the term count.

Lesson summary

A fixed difference defines an arithmetic sequence.
Term n is $a_1+(n-1)d$ .
The series sum is term count times the average of first and last.

Check your understanding

Question 1

For 2, 5, 8, ..., what is the fifth term?

14
17
11

Show answer and explanation

14

a_5=2+4(3)=14

.

Key terms

Common difference: The fixed amount added from one term to the next.
Series: The sum of terms of a sequence.

Continue through Math 20-1

View the complete Math 20-1 Alberta Grade 11 Mathematics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-1), expectation RF9. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

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RF9 · Analyze arithmetic sequences and series in context