Arithmetic series (adding the terms)

Covers AS91258

What a series is

A series is the sum of the terms of a sequence.
$S_n$ means the sum of the first $n$ terms (a partial sum).

The sum formula

For an arithmetic series there are two equivalent formulas:

$S_n = \frac{n}{2}\big(2a + (n - 1)d\big) \qquad \text{or} \qquad S_n = \frac{n}{2}(a + l)$

$n$ — how many terms you are adding.
$a$ — the first term; $l$ — the last term ( $u_n$ ).
$d$ — the common difference.
Use the first form when you know $a$ and $d$ ; use the second ( $\frac{n}{2}(a + l)$ ) when you already know the first and last terms.

Why the formula works

Pair the terms from the outside in: first $+$ last, second $+$ second-to-last, and so on. Every pair adds to the same total $(a + l)$ .
There are $\frac{n}{2}$ such pairs, giving $S_n = \frac{n}{2}(a + l)$ — this is the famous "Gauss" trick.

Using the sum formula

Decide which form fits what you are given.
Substitute and evaluate. If you need the last term first, get it from $u_n = a + (n-1)d$ .

Worked ExampleSumming a fixed number of terms

Find the sum of the first 20 terms of $4, 7, 10, 13, \ldots$

Step 1 — Identify $a$ , $d$ , $n$

$a = 4$ , $d = 3$ , $n = 20$ .

Step 2 — Substitute into the first form

$S_{20} = \frac{20}{2}\big(2(4) + (20 - 1)(3)\big) = 10\big(8 + 57\big)$

Step 3 — Evaluate

$S_{20} = 10 \times 65 = 650$

Worked ExampleSumming between known first and last terms

A stack of logs has 15 logs on the bottom row and 3 on the top, decreasing by 1 each row. How many logs are there in total?

Step 1 — Recognise the sequence

Rows: $15, 14, \ldots, 3$ — arithmetic with $a = 15$ , $l = 3$ . The number of rows is $n = 15 - 3 + 1 = 13$ .

Step 2 — Use the first-and-last form

$S_{13} = \frac{13}{2}(15 + 3) = \frac{13}{2}(18) = 13 \times 9 = 117$

There are 117 logs.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Find the sum of the first 15 terms of the sequence with $a = 6$ and $d = 4$ .

Merit

The first term of an arithmetic series is 7 and the 12th term is 51. Find the sum of the first 12 terms.

Excellence

A savings plan puts in $50 in month 1 and $8 more each following month. After how many months does the total saved first exceed $2000?

Arithmetic series (adding the terms)

Covers AS91258

What a series is

A series is the sum of the terms of a sequence.
$S_n$ means the sum of the first $n$ terms (a partial sum).

The sum formula

For an arithmetic series there are two equivalent formulas:

$S_n = \frac{n}{2}\big(2a + (n - 1)d\big) \qquad \text{or} \qquad S_n = \frac{n}{2}(a + l)$

$n$ — how many terms you are adding.
$a$ — the first term; $l$ — the last term ( $u_n$ ).
$d$ — the common difference.
Use the first form when you know $a$ and $d$ ; use the second ( $\frac{n}{2}(a + l)$ ) when you already know the first and last terms.

Why the formula works

Pair the terms from the outside in: first $+$ last, second $+$ second-to-last, and so on. Every pair adds to the same total $(a + l)$ .
There are $\frac{n}{2}$ such pairs, giving $S_n = \frac{n}{2}(a + l)$ — this is the famous "Gauss" trick.

Using the sum formula

Decide which form fits what you are given.
Substitute and evaluate. If you need the last term first, get it from $u_n = a + (n-1)d$ .

Worked ExampleSumming a fixed number of terms

Find the sum of the first 20 terms of $4, 7, 10, 13, \ldots$

Step 1 — Identify $a$ , $d$ , $n$

$a = 4$ , $d = 3$ , $n = 20$ .

Step 2 — Substitute into the first form

$S_{20} = \frac{20}{2}\big(2(4) + (20 - 1)(3)\big) = 10\big(8 + 57\big)$

Step 3 — Evaluate

$S_{20} = 10 \times 65 = 650$

Worked ExampleSumming between known first and last terms

A stack of logs has 15 logs on the bottom row and 3 on the top, decreasing by 1 each row. How many logs are there in total?

Step 1 — Recognise the sequence

Rows: $15, 14, \ldots, 3$ — arithmetic with $a = 15$ , $l = 3$ . The number of rows is $n = 15 - 3 + 1 = 13$ .

Step 2 — Use the first-and-last form

$S_{13} = \frac{13}{2}(15 + 3) = \frac{13}{2}(18) = 13 \times 9 = 117$

There are 117 logs.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Find the sum of the first 15 terms of the sequence with $a = 6$ and $d = 4$ .

Merit

The first term of an arithmetic series is 7 and the 12th term is 51. Find the sum of the first 12 terms.

Excellence

A savings plan puts in $50 in month 1 and $8 more each following month. After how many months does the total saved first exceed $2000?

Arithmetic series (adding the terms)

What a series is

The sum formula

Why the formula works

Using the sum formula

Step 1 — Identify aaa, ddd, nnn

Step 2 — Substitute into the first form

Step 3 — Evaluate

Step 1 — Recognise the sequence

Step 2 — Use the first-and-last form

Test yourself

Practice by grade

Arithmetic series (adding the terms)

What a series is

The sum formula

Why the formula works

Using the sum formula

Step 1 — Identify aaa, ddd, nnn

Step 2 — Substitute into the first form

Step 3 — Evaluate

Step 1 — Recognise the sequence

Step 2 — Use the first-and-last form

Test yourself

Practice by grade

Step 1 — Identify $a$ , $d$ , $n$

Step 1 — Identify $a$ , $d$ , $n$