Predicting What Comes Next: Exploring Sequences and Progressions · Lesson 2 of 8
Explicit Rules for Sequences
“An explicit rule skips the queue and jumps directly to any term you want.”
• Understand an explicit rule for a sequence. • Use n to calculate a term directly. • Find distant terms without generating earlier terms. • Determine whether a number belongs to a sequence. • Find the position of a term using an equation.
A number pattern such as 1, 3, 5, 7, … is easy to continue when we only need the next few terms. We can simply notice that each term increases by 2. But this method becomes impractical if we want to find a term much farther ahead, such as the 300th term.
Instead of writing every term one by one, we can look for a rule that connects the position of a term with its value. Such a rule allows us to calculate any term directly, no matter how far along the pattern it appears. This kind of rule is called an explicit rule, and it helps us describe patterns efficiently using algebra.
A formula that uses the position number n to calculate the value of the nth term directly.
Problem
Use uₙ=2n−1 to generate the first four odd numbers.
- 1.For n=1: u₁=2(1)−1=1.
- 2.For n=2: u₂=2(2)−1=3.
- 3.For n=3: u₃=2(3)−1=5.
- 4.For n=4: u₄=2(4)−1=7.
Why Explicit Rules Are Powerful
An explicit rule does not depend on earlier terms. If the rule is known, you can substitute n=20, n=53 or n=1000 directly.
Problem
For sₙ=5n−2, find the 100th term.
- 1.Substitute n=100.
- 2.s₁₀₀=5(100)−2.
- 3.s₁₀₀=498.
Is a Number in the Sequence?
An explicit rule can also answer the reverse question. Instead of giving n and finding the term, we can give the term and solve for n. The answer is valid only if n is a natural-number position.
Problem
Is 308 a term of the sequence sₙ=5n−2?
- 1.Set 5n−2=308.
- 2.Then 5n=310.
- 3.So n=62.
- 4.Since 62 is a natural number, 308 is the 62nd term.
Problem
Is 471 a term of sₙ=5n−2?
- 1.Set 5n−2=471.
- 2.Then 5n=473.
- 3.So n=94.6.
- 4.A sequence position must be a natural number, so 471 is not a term of this sequence.
Not Every Sequence Has an Obvious Simple Formula
Some sequences have very clear rules, such as odd numbers or square numbers. Others, such as the sequence of prime numbers, do not show a simple regular pattern that lets us predict the next term using a basic formula of the kind used here.
Practice Problems
- Find the first five terms of tₙ=4n−3.
- Find the 25th and 80th terms of uₙ=3n+2.
- Determine whether 167 is a term of tₙ=4n−1.
- Which term of the sequence sₙ=6n−5 is 301?
- Find an explicit rule for the sequence 2,5,8,11,… .
- Find an explicit rule for the square-number sequence.
Key Takeaways
• An explicit rule gives tₙ directly from n. • It allows us to jump to distant terms. • We can test sequence membership by solving tₙ=value. • The resulting n must be a natural-number position. • Some sequences have simple explicit rules; others do not.
Next, we describe sequences in a different way: by telling each term how to grow from earlier terms.