Sequences: sums, proofs and problems · Year 13
In Sequences: sums, proofs and problems (Year 13) you practise 8 skills: From one kind of rule to the other · Increasing, decreasing and negative terms · From the sum to a term · Proofs and counterexamples · Sequences inside a sequence · Everyday problems with arithmetic sequences · Everyday problems with geometric sequences · A sequence hidden in a recursive rule. The sum of the first terms is tied to the general term: the sum up to a position, minus the sum up to the position before, is the term in that position. The first term is the sum of one term. So a formula for the sum gives every term, and the difference or the ratio of the sequence.
To prove that a sequence is arithmetic, show that the difference between any term and the one before is the same number in every position, not only in the first three terms. To prove that it is geometric, show the same for the ratio. To show that a sequence is not arithmetic, or not geometric, one counterexample is enough: two differences, or two ratios, that are not equal. The terms in odd positions of an arithmetic sequence make an arithmetic sequence whose difference is twice the old one. The terms in even positions of a geometric sequence make a geometric sequence whose ratio is the old ratio squared. In an everyday problem, first decide which kind of sequence it is: a fixed amount added, like the seats in the rows of a hall, makes an arithmetic sequence, and a fixed multiplier, like growth by a percentage, makes a geometric one. Sometimes a recursive rule that multiplies and then subtracts hides a geometric sequence: take the same number away from every term, and each new term is a fixed multiple of the one before.
Tip: Before choosing a formula, check whether each term comes from the one before by adding a fixed amount or by multiplying by one.
What you learn
From one kind of rule to the other
For example: The general term of a sequence is aₙ = −6 + 6n . Which recursive rule defines the same sequence?
- a₁ = −6, aₙ₊₁ = aₙ + 6
- a₁ = 0, aₙ₊₁ = aₙ − 6
- a₁ = 6, aₙ₊₁ = aₙ − 6
- a₁ = 0, aₙ₊₁ = aₙ + 6
Answer and solution
The answer: a₁ = 0, aₙ₊₁ = aₙ + 6
Substitute n = 1 : a₁ = −6 + 6 = 0 . The difference is aₙ₊₁ − aₙ = 6 , so the rule is a₁ = 0, aₙ₊₁ = aₙ + 6 . Both forms give the first five terms 0, 6, 12, 18, 24 .
Increasing, decreasing and negative terms
For example: A geometric sequence has a₁ = −5 and q = 1 . Which is true of the sequence?
- The sequence is increasing
- The sequence is decreasing
- The sequence is constant
Answer and solution
The answer: The sequence is constant
The first terms are −5, −5, −5, −5 . With a ratio of one, every term is the same.
From the sum to a term
For example: The sum of the first n terms of a sequence is Sₙ = n² + 9n . What is the first term of the sequence?
Answer and solution
The answer: 10
The first term is the sum of one term. Substitute n = 1 : S₁ = 1² + 9 × 1 = 10 .
Proofs and counterexamples
For example: Which argument proves that the sequence aₙ = 3n + 9 is arithmetic?
- The first three differences are all 3
- a₂ − a₁ = 3
- The ratio aₙ₊₁ ÷ aₙ is the same for every n
- aₙ₊₁ − aₙ = 3 for every n
Answer and solution
The answer: aₙ₊₁ − aₙ = 3 for every n
Substitute n + 1 for n : aₙ₊₁ = 3(n + 1) + 9 . Subtracting aₙ , the constant cancels, so aₙ₊₁ − aₙ = 3 for every n . The difference does not depend on the position, so the sequence is arithmetic. Checking three differences only says nothing about the terms after them.
Sequences inside a sequence
For example: An arithmetic sequence has a₁ = −4 and d = 6 . The terms in odd positions, a₁, a₃, a₅ and so on, make a new sequence. What is the common difference of the new sequence?
Answer and solution
The answer: 12
The difference is 2 × 6 = 12 . For example, the new sequence starts −4 , 8 , and 8 − (−4) = 12 .
Everyday problems with arithmetic sequences
For example: A hall has 10 seats in its first row, and each row has 4 seats more than the row in front of it. How many seats are there in row 8?
Answer and solution
The answer: 38
Here a₁ = 10 and d = 4 , so a₈ = 10 + 7 × 4 = 38 . Row 8 has 38 seats.
Everyday problems with geometric sequences
For example: A tank holds 32 litres of water. Each hour half of the water in it flows out. How many litres are left in the tank after five hours?
Answer and solution
The answer: 1
The amounts are 32, 16, 8, 4, 2, 1 : after five hours there is one litre left.
A sequence hidden in a recursive rule
For example: A sequence is defined by a₁ = 3 and aₙ₊₁ = 3aₙ − 1 . What is a₃ ?
Answer and solution
The answer: 23
One after another: a₂ = 3 × 3 − 1 = 8 , a₃ = 3 × 8 − 1 = 23 .
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows selected AS and A level mathematics practice in England (the GCE subject content).