Sequences, Explicit & Recursive — Challenge
A sequence is a function whose inputs are the counting numbers. Describe it from the start, or from the term before.
What these problems cover
These questions use 7 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Ratio — A comparison of two quantities, like 3 cups flour to 2 cups water.
- Arithmetic sequence — A list that grows by adding the same amount each time.
- Domain — The set of allowed inputs.
- Sequence — An ordered list of numbers. The nth one is an.
- Explicit formula — A rule giving an directly from n.
- Recursive formula — A rule giving an from an-1, plus a starting value.
- Continuous — A function you could draw without lifting the pen — no holes, jumps or blow-ups.
Problem 1
An arithmetic sequence begins 3, 7, 11, …. What is its tenth term?
- 35
- 39
- 40
- 43
- 47
Worked answer
39
The common difference is 4, and the first term is 3.
Reaching the tenth term takes nine steps, not ten: 3+9· 4 = 3+36 = 39.
Using ten steps would give 43, which is the eleventh term.
Problem 2
What is the common difference of the sequence 5, 9, 13, …?
- 2
- 3
- 4
- 5
- 9
Worked answer
4
The common difference is what each term adds to the one before.
9-5 = 4 and 13-9 = 4, so the difference is 4 throughout.
Problem 3
A geometric sequence begins 2, 6, 18, …. What is its fifth term?
- 54
- 108
- 162
- 216
- 486
Worked answer
162
Each term is 3 times the one before, so the common ratio is 3.
The fifth term is four steps from the first: 2· 34 = 2· 81 = 162.
Listing them confirms it: 2, 6, 18, 54, 162.
Problem 4
What is the next term of 2, 4, 6, 8, …?
- 9
- 10
- 12
- 16
- 14
Worked answer
10
Each term is 2 more than the one before.
So the next term is 8+2 = 10.
Problem 5
What is the next term of 1, 4, 9, 16, …?
- 20
- 24
- 25
- 32
- 36
Worked answer
25
These are the square numbers: 12, 22, 32, 42.
The next is 52 = 25.
The gaps are 3, 5, 7, so the next gap is 9, giving 16+9 = 25 as well.
Problem 6
What is the next term of 5, 10, 20, 40, …?
- 50
- 60
- 70
- 80
- 100
Worked answer
80
Each term is double the one before, so the sequence is geometric with ratio 2.
The next term is 40· 2 = 80.
Adding the last difference instead would give 40+20 = 60, which misreads the pattern as arithmetic.
Problem 7
What is the next term of 100, 90, 80, …?
- 60
- 70
- 75
- 85
- 0
Worked answer
70
Each term is 10 less than the one before, so the common difference is -10.
The next term is 80-10 = 70.
Problem 8
A sequence has the rule 3n+1 for its nth term. What is its first term?
- 1
- 3
- 4
- 5
- 7
Worked answer
4
The first term means n=1.
3(1)+1 = 4.
The rule gives 4, 7, 10, …, an arithmetic sequence whose common difference is the 3 in the rule.