GCSE · Maths · AQA · Spec 8300 · Higher
Other sequences and surd ratios (Higher)
√2, 2, 2√2, 4, 4√2, … flips between surds and whole numbers, and its gaps keep changing. It looks like chaos, yet its rule is one of the simplest.
Sequences · Identify by testing
Which test does each sequence pass?
Pick a sequence and test it on paper first. Then put it in the column for the first test it passes. If it passes none of them, it goes in the last column.
Still to sort
Same first difference (0)
Subtract each term from the next one: every gap is the same.
Where the line is: An arithmetic sequence also has second differences of 0, but the first test has already settled it. Stop at the first test that works.
Same second difference (0)
The gaps change, but the gaps between the gaps are all the same (and not zero).
Where the line is: Changing gaps are not the end of the search: subtract the gaps again before you try anything else.
Same ratio (0)
Divide each term by the one before: the answer is the same every time.
Where the line is: When the differences never settle, divide. A constant ratio (a whole number, a fraction or a surd) means geometric.
Sum of the two before (0)
Each term is the two terms before it added together.
Where the line is: The pattern links three terms, not two, so neither a difference nor a ratio stays the same.
None of these tests (0)
No constant difference, second difference or ratio, and no sum of the two before.
Where the line is: Know named patterns such as the cube numbers on sight. For any other sequence, look for the rule you are given and apply it.
Run the four tests in order: subtract, subtract again, divide, look back two terms. The first one that works names the sequence.
Maths · Algebra
A surd ratio, line by line
Two ways a surd ratio turns up. Pick one, then step through the working.
The common ratio is any term divided by the term before it.
Step 1 of 6
The common ratio is any term divided by the term before it.
Using a rule you are given
Problem
Sequence A: the 1st term is 3, and each term after that is 3 × (previous term) − 4. Sequence B: the first two terms are 1 and 4, and each term after that is (previous term) + 2 × (the term before that). Find the next three terms of each sequence.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
Don't guess the pattern. Test it: subtract, subtract again, divide, look back two terms.
What you need to know
- Know the triangular numbers 1, 3, 6, 10, 15, …, the square numbers 1, 4, 9, 16, 25, … and the cube numbers 1, 8, 27, 64, 125, … on sight.
- Test an unfamiliar sequence in order: subtract neighbouring terms (same first difference: arithmetic), subtract again (same second difference: quadratic), divide each term by the one before (same ratio: geometric), then check whether each term is the sum of the two before (Fibonacci-type).
- A simple geometric progression rⁿ multiplies by the same number r each time. r can be a positive whole number or fraction, or a surd such as √2.
- To find a surd ratio, divide a term by the one before and simplify, rationalising the denominator if a surd is left there. Two terms two places apart divide to give r², so r is its square root.
- √a × √a = a, so multiplying twice by √a is the same as multiplying by a. That is why sequences such as √2, 2, 2√2, 4, … alternate between surds and whole numbers.
- When a question defines a term-to-term rule, substitute the previous term (or the two before) into it, never the position number, and feed each answer back in.
The big picture
You identify a sequence by testing how its terms are related: a constant first difference means arithmetic, a constant second difference means quadratic, a constant ratio means geometric, and a term made by adding the two before it means Fibonacci-type. This lesson runs those tests on familiar and unfamiliar sequences, works through a geometric progression whose common ratio is a surd, checks that method for the slip that spoils it, and applies term-to-term rules that a question gives you.
Key points
Worked example
Problem
The first five terms of a sequence are 4, 7, 12, 19, 28. Decide what kind of sequence it is, and find the 6th term.
⚠ Watch out
Stopping at the first test. When the gaps between terms are not equal, the sequence may still be quadratic (equal second differences), geometric (equal ratios) or Fibonacci-type, so subtract again, then divide, then look back two terms before deciding it has no simple rule.
Memory hook
Subtract, subtract again, divide, look back two. The messiest-looking sequence, √2, 2, 2√2, 4, …, is just × √2 every time.
Check yourself
Without looking back: what kind of sequence is 1, √11, 11, …, and what are its next two terms? Say which test proves it.
Flashcards
(9)What does a constant first difference tell you about a sequence?
The first differences change, but the second differences are all the same. What kind of sequence is it?
How do you test whether a sequence is geometric?
What makes a sequence Fibonacci-type?
Write down the first five cube numbers.
Why do the terms of a sequence with ratio √2 flip between surds and whole numbers?
How do you simplify a ratio such as 5 ÷ √5?
A geometric progression has positive terms. The 1st term is 2 and the 3rd term is 26. What is the common ratio?
A question says: next term = 2 × (previous term) + 3, and the 1st term is 4. What goes into the rule?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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