True or False? The common difference of an arithmetic sequence must be positive.

False. A negative common difference simply means each term is smaller than the one before. A common difference of zero is allowed too, and gives a sequence in which every term is the same.

How do you check whether a sequence is arithmetic?

Subtract each term from the one after it and see whether you always get the same number. If that difference changes anywhere at all, the sequence is not arithmetic.

A LevelMathsEdexcelPureFlashcardsSequences & SeriesArithmetic Sequences & Series

Arithmetic Sequences & Series (Edexcel A Level Maths: Pure): Flashcards

Exam code: 9MA0

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FrontArithmetic Sequences
Define common difference.

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Cards in this collection (10)

Define common difference.
The common difference $d$ is the fixed amount added to each term of an arithmetic sequence to get the next one.
It is the same between every pair of consecutive terms, which is what makes the sequence arithmetic.
The $n$ th term of an arithmetic sequence is:
$u_{n} =______+ (n -______) d$
$u_{n} = a + (n - 1) d$
Here $a$ is the first term, and the $n - 1$ appears because the first term has had no differences added to it yet.
You know the 4th and 9th terms of an arithmetic sequence. How do you find $a$ and $d$ ?
Write each term using $u_{n} = a + (n - 1) d$ to get two equations in $a$ and $d$ .
Subtracting one from the other eliminates $a$ and gives $d$ , and substituting back gives $a$ .
True or False?
The common difference of an arithmetic sequence must be positive.
False.
A negative common difference simply means each term is smaller than the one before.
A common difference of zero is allowed too, and gives a sequence in which every term is the same.
How do you check whether a sequence is arithmetic?
Subtract each term from the one after it and see whether you always get the same number.
If that difference changes anywhere at all, the sequence is not arithmetic.
The sum of the first $n$ terms of an arithmetic series is:
$S_{n} = \frac{n}{2} (2 a + (n -______) d)$ or $S_{n} = \frac{n}{2} (a +______)$
$S_{n} = \frac{n}{2} (2 a + (n - 1) d)$ or $S_{n} = \frac{n}{2} (a + l)$
Here $l$ is the last term of the series.
When is $S_{n} = \frac{n}{2} (a + l)$ the easier formula to use?
When you already know the last term of the series.
If you only have the first term and the common difference, the other form gets there in one step instead of two.
How is the formula for an arithmetic series proved?
Write the sum out once in order, then again in reverse, and add the two together.
Every pair of terms then adds to $2 a + (n - 1) d$ , and there are $n$ such pairs.
In the proof of the arithmetic series formula, why do you divide by $2$ at the end?
Because adding the two versions of the sum together gives twice the series, not the series itself.
Halving at the end undoes that doubling.
True or False?
$S_{n} = \frac{n}{2} (a + l)$ says the sum is $n$ times the average of the first and last terms.
True.
$\frac{n}{2} (a + l)$ is the same as $n \times \frac{a + l}{2}$ , and $\frac{a + l}{2}$ is exactly that average.
In an arithmetic series that is also the average of all the terms, which is why multiplying it by $n$ works.

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