Is it in the list?
Checking whether a number is a term
Is 137 in the list?
The odd numbers have the rule uₙ = 2n − 1. The book asks about the odd number 137: which position does it hold?
Is 137 a term of the sequence 1, 3, 5, 7, … ?
What this lesson covers
The idea
To check whether a number is a term of a sequence and find its position, set the explicit rule tₙ equal to it and solve for n; if n is not a natural number, it is not a term.
Is 137 in the list?
The odd numbers have the rule uₙ = 2n − 1. The book asks about the odd number 137: which position does it hold?
Is 137 a term of the sequence 1, 3, 5, 7, … ?
- Yes, it is an odd number
- No
- I cannot tell without writing all the terms
Solve for n
The rule is sₙ = 5n − 2. Test a number: the toy sets 5n − 2 equal to it and solves for n.
n must be a natural number
To check whether a number is a term, set the explicit rule tₙ equal to it and solve for n. If n is a natural number, the number is a term and n is its position. If n is not a natural number, it is not a term.
Book example: 2n − 1 = 137 gives n = 69, so 137 is the 69th term of the odd numbers. And 5n − 2 = 308 gives 5n = 310, n = 62: the 62nd term.
Notes
To check whether a number is a term, set tₙ equal to it and solve for n. If n is a natural number, it is a term (the nth); if not, it is not a term.
Check yourself
Which position does 99 hold in the odd numbers, uₙ = 2n − 1? (n = ?)
Answer: 50
2n − 1 = 99, so 2n = 100 and n = 50. 99 is the 50th odd number.
Is 213 a term of sₙ = 5n − 2?
Is 50 a term of tₙ = 3n − 2? (The sequence is 1, 4, 7, 10, …)
In sₙ = 5n − 2, which position holds the term 473? (n = ?)
Answer: 95
5n − 2 = 473, so 5n = 475 and n = 95.
- No, because n = 42.2. You subtracted 2 instead of adding it. From 5n − 2 = 213 we get 5n = 213 + 2 = 215, and n = 43.
- Yes, it is the 43rd term — correct. 5n − 2 = 213, so 5n = 215 and n = 43, a natural number.
- Yes, it is the 42nd term. Check: 5 × 42 − 2 = 208, not 213. Solving gives n = 43.
- Yes, it is the 17th term. Rounding is not allowed. 3n − 2 = 50 gives 3n = 52, so n is about 17.3, which is not a natural number.
- Yes, it is the 18th term. Check: 3 × 18 − 2 = 52, not 50. And n must come out as an exact natural number.
- No, because n is not a natural number — correct. 3n = 52, so n is about 17.3. That is not a natural number, so 50 is not a term.