a22 finding the nth term
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A22 Finding the nth term
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Sequences of multiples
All sequences of multiples can be generated by adding the same amount each time. They are linear sequences.
For example, the sequence of multiples of 5:
5, 10, 15, 20, 25, 30, 35, 40, …
+5 +5 +5 +5 +5 +5 +5
can be found by adding 5 each time.
Compare the terms in the sequence of multiples of 5 to their position in the sequence:
Position
Term
1
5
2
10
3
15
4
20
5
25
n…
…
× 5 × 5 × 5 × 5 × 5 × 5
5n
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Sequences of multiples
The sequence of multiples of 3:3, 6, 9, 12, 15, 18, 21, 24, …
+3 +3 +3 +3 +3 +3 +3
can be found by adding 3 each time.
Compare the terms in the sequence of multiples of 3 to their position in the sequence:
Position
Term
1
3
2
6
3
9
4
12
5
15
n…
…×3 ×3 ×3 ×3 ×3 ×3
3n
The nth term of a sequence of multiples is always dn, where d is the difference between consecutive terms.
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Sequences of multiples
The nth term of a sequence of multiples is always dn, where d is the difference between consecutive terms.
For example:
The nth term of 4, 8, 12, 16, 20, 24, … is 4n
The 10th term of this sequence is 4 × 10 = 40
The 25th term of this sequence is 4 × 25 = 100
The 47th term of this sequence is 4 × 47 = 188
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Finding the nth term of a linear sequence
The terms in this sequence4, 7, 10, 13, 16, 19, 22, 25, …
+3 +3 +3 +3 +3 +3 +3
can be found by adding 3 each time.
Compare the terms in the sequence to the multiples of 3.
Position
Multiples of 3
1 2 3 4 5 n…× 3 × 3 × 3 × 3 × 3 × 3
3n
Term 4 7 10 13 16 …
3 6 9 12 15+ 1 + 1 + 1 + 1 + 1 + 1
3n + 1
Each term is one more than a multiple of 3.
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Finding the nth term of a linear sequence
The terms in this sequence1, 6, 11, 16, 21, 26, 31, 36, …
+5 +5 +5 +5 +5 +5 +5
can be found by adding 5 each time.
Compare the terms in the sequence to the multiples of 5.
Position
Multiples of 5
1 2 3 4 5 n…× 5 × 5 × 5 × 5 × 5 × 5
5n
Term 1 6 11 16 21 …
5 10 15 20 25– 4 – 4 – 4 – 4 – 4 – 4
5n – 4
Each term is four less than a multiple of 5.
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Finding the nth term of a linear sequence
The terms in this sequence5, 3, 1, –1, –3, –5, –7, –9, …
–2 –2 –2 –2 –2 –2 –2
can be found by subtracting 2 each time.
Compare the terms in the sequence to the multiples of –2.
Position
Multiples of –2
1 2 3 4 5 n…× –2 × –2 × –2 × –2 × –2 × –2
–2n
Term 5 3 1 –1 –3 …
–2 –4 –6 –8 –10+ 7 + 7 + 7 + 7 + 7 + 7
7 – 2n
Each term is seven more than a multiple of –2.
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Sequences that increase (or decrease) in equal steps are called linear or arithmetic sequences.
The difference between any two consecutive terms in an arithmetic sequence is a constant number.
When we describe arithmetic sequences we call the difference between consecutive terms, d.
We call the first term in an arithmetic sequence, a.
If an arithmetic sequence has a = 5 and d = –2,
We have the sequence:
5, 3, 1, –1, –3, –5, ...
Arithmetic sequences
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The rule for the nth term of any arithmetic sequence is of the form: T(n) = an + b
a and b can be any number, including fractions and negative numbers.For example:
T(n) = 2n + 1 Generates odd numbers starting at 3.
T(n) = 2n + 4 Generates even numbers starting at 6.
T(n) = 2n – 4 Generates even numbers starting at –2.
T(n) = 3n + 6 Generates multiples of 3 starting at 9.
T(n) = 4 – n Generates descending integers starting at 3.
The nth term of an arithmetic sequence