Arithmetic Sequences and Series

In short: An arithmetic sequence adds the same amount $d$ each time, like 3, 7, 11, 15. Its $n$ th term is $T_{n} = a + (n - 1) d$ , and the sum of its first $n$ terms is $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ .

Starting point: Years 9 and 10

Number patterns

A sequence is a list of numbers in order. Each number is a term. The first term is $T_{1}$ , the second is $T_{2}$ , and so on.

To continue a pattern, find the rule. In 4, 7, 10, 13, ..., each term is 3 more than the last.

Finding a rule for the nth term

If the difference is always 3, the rule has $3 n$ in it. Compare:

$n$	1	2	3	4
$3 n$	3	6	9	12
term	4	7	10	13

Each term is one more than $3 n$ , so $T_{n} = 3 n + 1$ . Then $T_{50} = 151$ .

Patterns and straight lines

Plot $n$ against $T_{n}$ and the points lie on a straight line. The common difference is the gradient. That is why arithmetic sequences are sometimes called linear sequences.

Arithmetic sequences

Common difference d

Same amount added
d = T₂ - T₁

General term

Tₙ = a + (n - 1)d
Linear pattern

Series

Sum of terms
Sₙ = n/2 (a + l)
Sₙ = n/2 (2a + (n-1)d)

Sigma notation

Σ means add up

Year 12Arithmetic series

A series is the sum of the terms of a sequence. Write the sum forwards and backwards and add: every pair adds to $a + l$ , where $l$ is the last term. So

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

Isla's jar after 20 weeks: $S_{20} = \frac{20}{2} (5 + 62) = 670$ dollars.

Sigma notation. $k = 1 \sum n T_{k}$ means "add $T_{k}$ for $k = 1$ to $n$ ."

$k = 1 \sum 4 (2 k + 1) = 3 + 5 + 7 + 9 = 24$

How many terms? How many terms of $3 + 7 + 11 + \dots$ add to 300? Solve $\frac{n}{2} [6 + 4 (n - 1)] = 300$ , which is $2 n^{2} + n - 300 = 0$ , giving $n = 12$ (ignore the negative root).

A term from the sum formula. If $S_{n}$ is given, then $T_{n} = S_{n} - S_{n - 1}$ , because the sum of $n$ terms minus the sum of $n - 1$ terms leaves the last term. For $S_{n} = n^{2} + 4 n$ : $T_{10} = S_{10} - S_{9} = 140 - 117 = 23$ .

Inequalities in $n$ . For "the first row with more than 100 seats" where $T_{n} = 12 + 2 (n - 1)$ , solve $12 + 2 (n - 1) > 100$ : $n > 45$ , so row 46. For a sum, solve the quadratic $S_{n} =$ target, then take the next whole number. Check the answer and the one before it.

Splitting a sum. $(2 + 1) + (4 + 1) + (8 + 1) + \dots$ for 10 terms is a geometric series $2 + 4 + 8 + \dots$ plus ten 1s: $\frac{2 ( 2 ^{10} - 1 )}{2 - 1} + 10 = 2046 + 10 = 2056$ .

Quick summary

An arithmetic sequence adds the same amount $d$ each step.
$T_{n} = a + (n - 1) d$ , where $a$ is the first term.
The graph of $T_{n}$ against $n$ is a straight line with gradient $d$ .
$S_{n} = \frac{n}{2} (a + l) = \frac{n}{2} [2 a + (n - 1) d]$ .
$\sum$ means add up the terms.

Quiz

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What is the next term of 7, 11, 15, 19, ...?
1. A 22
2. B 23
3. C 24
4. D 21
Show answer
B: 23
Add 4 each time.
What is the common difference of 20, 17, 14, 11, ...?
1. A 3
2. B $- 3$
3. C 17
4. D $- 17$
Show answer
B: $- 3$
Each term is 3 less.
A pattern has rule $T_{n} = 5 n - 2$ . What is $T_{6}$ ?
1. A 30
2. B 28
3. C 32
4. D 23
Show answer
B: 28
$5 (6) - 2 = 28$ .