A progression , also called a sequence , is an ordered list of numbers together with a rule for finding them all. The individual numbers in it are called terms .

You are given a list of numbers. How do you decide whether it is arithmetic or geometric?

Test the differences between consecutive terms, then test the ratios of consecutive terms. If every difference is the same it is arithmetic, if every ratio is the same it is geometric, and if neither is constant it is neither.

A quantity changes repeatedly. How can you tell whether to model it with an arithmetic progression or a geometric one?

Use an arithmetic progression when a fixed amount is added or subtracted each time. Use a geometric progression when the quantity is multiplied by a fixed amount each time, which includes changing by a fixed percentage.

True or False? If a quantity increases by the same percentage every year, it increases by the same amount every year.

False. The percentage stays the same, but it is applied to a larger amount each time, so the actual increase grows. A profit of £20 000 rising by 10% gains £2 000 in the first year and £2 200 in the second.

A LevelMathsCambridge (CIE)Pure 1FlashcardsSeriesSequences & Series

Sequences & Series (Cambridge (CIE) A Level Maths: Pure 1): Flashcards

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Define progression.

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Define progression.
A progression, also called a sequence, is an ordered list of numbers together with a rule for finding them all.
The individual numbers in it are called terms.
What is the difference between a progression and a series?
A progression is the list of terms itself, while a series is what you get by adding those terms together.
So $1, 4, 7, 10, 13$ is a progression and $1 + 4 + 7 + 10 + 13$ is the series that goes with it.
In the progression $1, 4, 7, 10, 13, \dots$ complete both values:
$u_{4} =______and S_{4} =______$
The completed values are:
$u_{4} = 10 and S_{4} = 22$
The notation $u_{n}$ means the $n$ th term on its own, while $S_{n}$ means the sum of the first $n$ terms, here $1 + 4 + 7 + 10$ .
You are given a list of numbers. How do you decide whether it is arithmetic or geometric?
Test the differences between consecutive terms, then test the ratios of consecutive terms.
If every difference is the same it is arithmetic, if every ratio is the same it is geometric, and if neither is constant it is neither.
Complete the two conditions on three numbers $a$ , $b$ and $c$ , filling in the missing expression in $b$ each time:
$______= a + c (arithmetic progression)$
$______= a c (geometric progression)$
The completed conditions are:
$2 b = a + c (arithmetic progression)$
$b^{2} = a c (geometric progression)$
Both come from there being a single common difference or a single common ratio, so the middle term is pinned down by the two outside it.
True or False?
In the notation $u_{5}$ , the $5$ tells you the value of that term.
False.
The subscript gives the position of the term, not its value.
In the progression $1, 4, 7, 10, 13, \dots$ the fifth term is $u_{5} = 13$ , so the subscript and the value are different numbers.
A quantity changes repeatedly. How can you tell whether to model it with an arithmetic progression or a geometric one?
Use an arithmetic progression when a fixed amount is added or subtracted each time.
Use a geometric progression when the quantity is multiplied by a fixed amount each time, which includes changing by a fixed percentage.
Complete the common ratio for each of these geometric models:
A quantity that increases by 10% each year has $r =______$ .
A quantity that decreases by 5% each year has $r =______$ .
The completed ratios are:
A quantity that increases by 10% each year has $r = 1.1$ .
A quantity that decreases by 5% each year has $r = 0.95$ .
Each term is the whole of the previous one after the change, not just the change itself.
True or False?
If a quantity increases by the same percentage every year, it increases by the same amount every year.
False.
The percentage stays the same, but it is applied to a larger amount each time, so the actual increase grows.
A profit of £20 000 rising by 10% gains £2 000 in the first year and £2 200 in the second.
A grandfather gives £10 on a first birthday and doubles the gift each year. Which formula gives the gift on the 18th birthday, and which gives the total given by then?
The gift on the 18th birthday is a single term, so use the $n$ th term formula for a geometric progression with $n = 18$ .
The total given by then adds up all 18 gifts, so use the sum formula instead.
Someone pays money into an account every month from January to December. When you model the year's total as an arithmetic series, what is $n$ ?
There are $12$ terms.
There is one payment in each of the twelve months and the January payment is the first term, so twelve payments are added in all.
In a savings plan modelled as an arithmetic progression, what does the common difference represent?
The amount by which each payment is bigger than the one before.
It is not the size of a payment itself, which starts at the first term $a$ and grows from there.

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