Arithmetic sums | Year 11 | Ascent Library

01 / LEARN

The idea, step by step

Pairing forwards and backwards

A series is a sum of sequence terms. Write the arithmetic list forwards and backwards. Each paired column contains the same total, first term plus last term, and there are $n$ columns. This counts the original sum twice.

The derivation also works for negative differences and negative terms. Retain signs instead of adding absolute values.

S_{n} = \frac{n (a + u_{n})}{2} = \frac{n}{2} [2 a + (n - 1) d]

Counting terms before summing

Use the first-and-last form when both endpoints are known. If needed, first solve the term equation to determine how many terms occur. An inclusive interval of integers from $p$ to $q$ contains $q - p + 1$ terms.

Finding $n$ from a sum can produce a quadratic. Keep only positive-integer roots and check whether the resulting terms make sense in context.

A common trap

Multiply the number of terms by the first term.

Instead: The terms change; multiply by the mean of the first and last terms.

02 / WORKED EXAMPLES

See the thinking

Example 1

A sum with known endpoints

Find $12 + 17 + 22 + \dots + 97$ .

$97 = 12 + 5 (n - 1)$ gives $n = 18$ .
$S = 18 (12 + 97) / 2 = 981$ .

981

Example 2

Recovering the row count

Rows contain $8, 11, 14, \dots$ seats. The total is 215. Find the number of rows.

$215 = n (3 n + 13) / 2$ gives $3 n^{2} + 13 n - 430 = 0$ .
$(n - 10) (3 n + 43) = 0$ . Only $n = 10$ is valid.

10

Check your understanding

Find the sum of the first 20 positive even integers.

Show answer

$420$ .

The endpoints are 2 and 40; $20 (2 + 40) / 2 = 420$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find $S_{15}$ when $a = 3, d = 4$ .
2 marks
Find $100 + 94 + 88 + \dots + 4$ .
2 marks
Find the sum of the integers from 23 to 68 inclusive.
2 marks
A 12-term arithmetic series starts at 7 and sums to 282. Find its last term and difference.
2 marks
A runner begins at 2 km on day 1 and adds 0.5 km each day. Find the distance over 14 days.
2 marks
Find positive integer $n$ if $1 + 4 + \dots + (3 n - 2) = 145$ .
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find $S_{15}$ when $a = 3, d = 4$ .
2 marks
$15 [6 + 14 (4)] / 2 = 465$ .
$465$ .
Find $100 + 94 + 88 + \dots + 4$ .
2 marks
$4 = 100 - 6 (n - 1)$ gives $n = 17$ ; $17 (104) / 2 = 884$ .
$884$ .
Find the sum of the integers from 23 to 68 inclusive.
2 marks
There are 46 terms; $46 (23 + 68) / 2 = 2093$ .
$2093$ .
A 12-term arithmetic series starts at 7 and sums to 282. Find its last term and difference.
2 marks
$282 = 6 (7 + l)$ gives $l = 40$ ; $(40 - 7) / 11 = 3$ .
Last term 40; $d = 3$ .
A runner begins at 2 km on day 1 and adds 0.5 km each day. Find the distance over 14 days.
2 marks
$S = 14 [4 + 13 (0.5)] / 2 = 73.5$ .
$73.5 km$ .
Find positive integer $n$ if $1 + 4 + \dots + (3 n - 2) = 145$ .
3 marks
$n (3 n - 1) / 2 = 145$ ; $(n - 10) (3 n + 29) = 0$ .
$10$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics sequence and series content with introductory induction, regrouped into an original school-sized chapter.

Official curriculum source ↗

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