Define an arithmetic sequence .

An arithmetic sequence is one whose terms change by the same amount each time , and that amount is called the common difference . Arithmetic sequences are also called linear sequences.

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Sequences (Edexcel IGCSE Maths B): Flashcards

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FrontIntroduction to Sequences
Fill in the two missing words.
Each number in a sequence is called a $______$ and where it sits in the sequence is called its $______$ number.

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Fill in the two missing words.
Each number in a sequence is called a $______$ and where it sits in the sequence is called its $______$ number.
The completed sentence is:
Each number in a sequence is called a term and where it sits in the sequence is called its position number.
The letter $n$ is used for position, so $n = 1$ gives the first term, and the subscript form $a_{n}$ means the $n$ th term.
Define a term-to-term rule and a position-to-term rule.
A term-to-term rule tells you how to get from one term to the next, such as 'add 10 each time'.
A position-to-term rule is an expression in $n$ , also called the $n$ th term formula, which gives any term directly from its position.
A sequence starts at 4 and its rule is to add 10 each time. Write down the first four terms.
The first four terms are $4$ , $14$ , $24$ and $34$ .
A term-to-term rule is applied to the term you are on, so each term is 10 more than the one before it.
Why does an $n$ th term formula let you jump straight to the 100th term?
Because you substitute $n = 100$ directly into the formula, without needing to work out any of the earlier terms.
For the formula $8 n + 2$ the 100th term is $8 \times 100 + 2 = 802$ .
A sequence has $n$ th term $8 n + 2$ . How do you decide whether 98 is in the sequence?
Set the formula equal to the value and solve for $n$ , so $8 n + 2 = 98$ gives $n = 12$ .
Because $n$ comes out as a whole number, 98 is in the sequence, and it is the 12th term.
True or False?
Solving $8 n + 2 = 124$ gives $n = 15.25$ , so 124 is the 15th term of that sequence.
False.
A position has to be a whole number, so $n = 15.25$ means 124 is not in the sequence at all.
Rounding to the nearest term is not an option: a value either fits the formula exactly or it does not belong.
Define an arithmetic sequence.
An arithmetic sequence is one whose terms change by the same amount each time, and that amount is called the common difference.
Arithmetic sequences are also called linear sequences.
True or False?
The sequence 11, 8, 5, 2, $- 1$ , ... is an arithmetic sequence.
True.
Each term is 3 less than the one before, which is a common difference of $- 3$ .
A common difference may be negative, so an arithmetic sequence can decrease as well as increase.
Complete the formula for the $n$ th term of an arithmetic sequence.
$u_{n} = a + (n -______) d$
The completed formula is:
$u_{n} = a + (n - 1) d$
Here $a$ is the first term and $d$ is the common difference.
Why does the arithmetic sequence formula use $n - 1$ rather than $n$ ?
The first term is $a$ on its own, so when $n = 1$ no common difference has been added yet.
By the $n$ th term the difference has been added $n - 1$ times, and that is what the formula counts.
An arithmetic sequence has $a = 10$ and $d = 4$ . What is its third term?
Substituting $n = 3$ into $u_{n} = a + (n - 1) d$ gives $u_{3} = 10 + (3 - 1) \times 4 = 18$ .
Leaving $n$ in instead gives an expression for any term, $u_{n} = 10 + 4 (n - 1) = 4 n + 6$ .
The 6th term of an arithmetic sequence is 13 and the 21st term is 43. How do you find $a$ and $d$ ?
Substitute each term into $u_{n} = a + (n - 1) d$ to get two equations, $a + 5 d = 13$ and $a + 20 d = 43$ .
Solving those simultaneously gives $d = 2$ , and substituting back gives $a = 3$ .

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