Explicit Rules

An explicit rule is a formula that allows us to calculate any term in a sequence directly from its position.

Unlike a recursive rule, we do not need to know the previous term.

For example, consider:

5, 8, 11, 14, 17, ...

An explicit rule for this sequence is:

aₙ = 3n + 2

where:

• aₙ = value of the nth term
• n = position or term number

If we want the 20th term, we substitute n = 20:

a₂₀ = 3(20) + 2

a₂₀ = 60 + 2

a₂₀ = 62

We can find the 20th term without calculating the first 19 terms.

What Makes a Rule Explicit?

An explicit rule connects the term number directly to the term value.

Consider:

4, 9, 14, 19, 24, ...

An explicit rule is:

aₙ = 5n − 1

To find a term, simply substitute its position.

For the 6th term:

a₆ = 5(6) − 1

a₆ = 30 − 1

a₆ = 29

For the 100th term:

a₁₀₀ = 5(100) − 1

a₁₀₀ = 500 − 1

a₁₀₀ = 499

This is the main advantage of an explicit rule: distant terms can be calculated immediately.

Understanding the Notation

In:

aₙ = 4n + 3

the symbol n represents the term number.

The symbol aₙ represents the value at that position.

For example:

If:

aₙ = 4n + 3

then:

a₁ = 4(1) + 3 = 7

a₂ = 4(2) + 3 = 11

a₃ = 4(3) + 3 = 15

So the sequence is:

7, 11, 15, 19, ...

Generating a Sequence from an Explicit Rule

Suppose:

aₙ = 6n − 2

Generate the first five terms.

For n = 1:

a₁ = 6(1) − 2 = 4

For n = 2:

a₂ = 6(2) − 2 = 10

For n = 3:

a₃ = 6(3) − 2 = 16

For n = 4:

a₄ = 6(4) − 2 = 22

For n = 5:

a₅ = 6(5) − 2 = 28

Therefore:

4, 10, 16, 22, 28, ...

Substitution

Using explicit rules requires careful substitution.

Suppose:

aₙ = 7n + 4

Find a₁₂.

Because we want the 12th term:

n = 12

Substitute:

a₁₂ = 7(12) + 4

a₁₂ = 84 + 4

a₁₂ = 88

A useful habit is to write the substitution step before calculating.

Finding Distant Terms

Consider:

aₙ = 8n − 5

Find the 200th term.

Substitute:

n = 200

a₂₀₀ = 8(200) − 5

a₂₀₀ = 1600 − 5

a₂₀₀ = 1595

We did not need to calculate terms 1 through 199.

This makes explicit rules particularly useful when dealing with large term numbers.

Explicit Rules for Arithmetic Sequences

An arithmetic sequence has a constant difference between consecutive terms.

For example:

7, 12, 17, 22, 27, ...

The common difference is:

d = 5

For an arithmetic sequence, the explicit rule can be written:

aₙ = a₁ + (n − 1)d

where:

• aₙ = nth term
• a₁ = first term
• n = term number
• d = common difference

Using the Arithmetic Sequence Formula

For:

7, 12, 17, 22, ...

we have:

a₁ = 7

d = 5

Therefore:

aₙ = 7 + (n − 1)(5)

Simplify:

aₙ = 7 + 5n − 5

aₙ = 5n + 2

Now any term can be calculated directly.

For example:

a₅₀ = 5(50) + 2

a₅₀ = 252

Finding an Explicit Rule from a Sequence

Consider:

8, 11, 14, 17, 20, ...

First identify the common difference:

d = 3

Start with:

3n

This produces:

3, 6, 9, 12, 15, ...

Compare this with the actual sequence:

8, 11, 14, 17, 20, ...

Each actual term is 5 greater.

Therefore:

aₙ = 3n + 5

Check:

a₁ = 3(1) + 5 = 8 ✓

a₂ = 3(2) + 5 = 11 ✓

a₃ = 3(3) + 5 = 14 ✓

Another Example

Find an explicit rule for:

14, 20, 26, 32, 38, ...

Common difference:

d = 6

Start with:

6n

This gives:

6, 12, 18, 24, 30, ...

The actual sequence is always 8 greater.

Therefore:

aₙ = 6n + 8

Check the first term:

a₁ = 6(1) + 8 = 14

The rule works.

Explicit Rules for Decreasing Sequences

Explicit rules can also describe decreasing patterns.

Consider:

30, 26, 22, 18, 14, ...

The common difference is:

−4

Use:

aₙ = a₁ + (n − 1)d

Substitute:

aₙ = 30 + (n − 1)(−4)

Simplify:

aₙ = 30 − 4n + 4

aₙ = 34 − 4n

Check:

a₁ = 34 − 4(1) = 30

a₂ = 34 − 4(2) = 26

The rule works.

Finding the Term Number

Explicit rules can also be used in reverse.

Suppose:

aₙ = 5n + 2

Which term has a value of 72?

Set:

aₙ = 72

Therefore:

72 = 5n + 2

Subtract 2:

70 = 5n

Divide by 5:

n = 14

Therefore, 72 is the 14th term.

Is a Number in the Sequence?

Suppose:

aₙ = 4n + 1

Is 101 in the sequence?

Set:

101 = 4n + 1

100 = 4n

n = 25

Since 25 is a positive whole-number term position:

Yes, 101 is the 25th term.

Now ask whether 100 is in the sequence:

100 = 4n + 1

99 = 4n

n = 24.75

A term number must be a positive whole number.

Therefore:

100 is not a term in the sequence.

Recursive Rules

Recall that a recursive rule defines each term using the previous term.

For:

5, 8, 11, 14, 17, ...

a recursive rule is:

a₁ = 5

aₙ = aₙ₋₁ + 3

To generate later terms, we repeatedly add 3.

Explicit Rule for the Same Sequence

The same sequence can be written explicitly as:

aₙ = 3n + 2

Now we can calculate any term directly.

For example:

a₁₀₀ = 3(100) + 2

a₁₀₀ = 302

This is much faster than applying the recursive rule 99 times.

Comparing Explicit and Recursive Rules

Consider:

10, 14, 18, 22, 26, ...

Recursive

a₁ = 10

aₙ = aₙ₋₁ + 4

This tells us:

Start at 10 and repeatedly add 4.

Explicit

aₙ = 4n + 6

This tells us:

Multiply the term number by 4 and add 6.

Both rules generate exactly the same sequence.

Why Both Forms Are Useful

Recursive rules are often easier to identify because we naturally look at how consecutive terms change.

For example:

20, 27, 34, 41, ...

It is easy to see:

add 7

So:

a₁ = 20

aₙ = aₙ₋₁ + 7

But if we want the 500th term, an explicit rule is much more efficient:

aₙ = 7n + 13

Then:

a₅₀₀ = 7(500) + 13

a₅₀₀ = 3513

Converting a Recursive Arithmetic Rule to Explicit Form

Suppose:

a₁ = 9

aₙ = aₙ₋₁ + 4

We know:

a₁ = 9

d = 4

Use:

aₙ = a₁ + (n − 1)d

Therefore:

aₙ = 9 + 4(n − 1)

aₙ = 9 + 4n − 4

aₙ = 4n + 5

So the recursive and explicit forms describe the same sequence.

Explicit Rules Beyond Arithmetic Sequences

Not every explicit rule is linear.

Consider:

1, 4, 9, 16, 25, ...

These are square numbers.

The explicit rule is:

aₙ = n²

Therefore:

a₁₀ = 10² = 100

a₅₀ = 50² = 2500

Another example:

2, 5, 10, 17, 26, ...

This follows:

aₙ = n² + 1

because:

1² + 1 = 2

2² + 1 = 5

3² + 1 = 10

4² + 1 = 17

Explicit Rules for Geometric Sequences

A sequence that repeatedly multiplies by the same factor is called a geometric sequence.

For example:

3, 6, 12, 24, 48, ...

The recursive rule is:

a₁ = 3

aₙ = 2aₙ₋₁

An explicit rule is:

aₙ = 3(2)ⁿ⁻¹

Check the fourth term:

a₄ = 3(2)⁴⁻¹

a₄ = 3(2)³

a₄ = 3(8)

a₄ = 24

General Geometric Formula

For a geometric sequence:

aₙ = a₁rⁿ⁻¹

where:

• a₁ = first term
• r = common ratio
• n = term number

Example

Consider:

5, 15, 45, 135, ...

Here:

a₁ = 5

r = 3

Therefore:

aₙ = 5(3)ⁿ⁻¹

Find the 8th term:

a₈ = 5(3)⁷

a₈ = 5(2187)

a₈ = 10,935

The explicit formula allows us to jump directly to the eighth term.

Real-World Example: Saving Money

Suppose you begin with $50 and add $20 each week.

The amounts are:

50, 70, 90, 110, 130, ...

An explicit rule is:

aₙ = 50 + 20(n − 1)

or:

aₙ = 20n + 30

After 30 weeks:

a₃₀ = 20(30) + 30

a₃₀ = $630

Real-World Example: Doubling

Suppose a bacteria population begins with 100 bacteria and doubles during each time interval.

The sequence is:

100, 200, 400, 800, 1600, ...

An explicit rule is:

aₙ = 100(2)ⁿ⁻¹

This model allows us to calculate any stage directly.

Choosing Between Explicit and Recursive Rules

Use a recursive rule when you want to emphasize:

• how one term changes into the next
• a repeated process
• consecutive terms

Use an explicit rule when you want to:

• find a distant term
• connect term position directly to value
• avoid calculating every previous term
• determine whether a particular value appears in a sequence

In many situations, both forms are useful.

A Strategy for Using Explicit Rules

When given an explicit formula:

First: identify the requested term number.

Next: set n equal to that term number.

Then: substitute n into the formula.

Next: follow the order of operations carefully.

Finally: check whether the result is reasonable for the sequence.

For example:

aₙ = 6n − 1

Find a₂₅.

n = 25

a₂₅ = 6(25) − 1

a₂₅ = 150 − 1

a₂₅ = 149

Common Mistakes

Using the term value instead of the term number

If you are finding the 20th term, substitute:

n = 20

Starting with n = 0 when the sequence begins at n = 1

Unless otherwise stated, sequences in these notes begin with:

n = 1

Confusing explicit and recursive rules

aₙ = 4n + 3 is explicit.

aₙ = aₙ₋₁ + 4 is recursive.

Forgetting the (n − 1) in arithmetic formulas

The arithmetic formula is:

aₙ = a₁ + (n − 1)d

not:

aₙ = a₁ + nd

Forgetting the (n − 1) exponent in geometric formulas

The standard geometric formula is:

aₙ = a₁rⁿ⁻¹

because the first term has undergone zero multiplications by r.

Listing every previous term unnecessarily

If an explicit rule is available, use it to calculate the required term directly.

Key Terms

Explicit rule — a formula that calculates a term directly from its position.

Sequence — an ordered list of terms.

Term — an individual value in a sequence.

Term number — the position of a term in a sequence.

nth term — the term at position n.

aₙ — notation representing the nth term.

Substitution — replacing a variable with a known value.

Arithmetic sequence — a sequence with a constant difference between consecutive terms.

Common difference (d) — the constant amount added or subtracted in an arithmetic sequence.

Geometric sequence — a sequence in which each term is obtained by multiplying the previous term by a constant factor.

Common ratio (r) — the constant multiplication factor in a geometric sequence.

Recursive rule — a rule that defines terms using one or more previous terms.

Key Takeaways

• An explicit rule calculates a term directly from its term number.
• The variable n represents the position of a term.
• aₙ represents the value of the nth term.
• Explicit rules allow distant terms to be calculated without listing earlier terms.
• For arithmetic sequences: aₙ = a₁ + (n − 1)d.
• For geometric sequences: aₙ = a₁rⁿ⁻¹.
• Explicit rules can also describe patterns such as aₙ = n².
• Recursive rules use previous terms; explicit rules use the term number.
• The same sequence can often be represented both recursively and explicitly.
• When using an explicit formula, identify n, substitute carefully, and follow the order of operations.
• Explicit formulas are especially useful for finding distant terms such as the 50th, 100th, or 1000th term.

Check Your Understanding

1. Explain what makes a rule explicit.
2. Given aₙ = 4n + 3, find a₁, a₂, a₃, and a₄.
3. Given aₙ = 7n − 2, find the 20th term.
4. Given aₙ = 5n + 8, find the 100th term.

5. Find an explicit rule for:

   6, 10, 14, 18, 22, ...

6. Find an explicit rule for:

   15, 22, 29, 36, 43, ...

7. Find an explicit rule for:

   40, 35, 30, 25, 20, ...

8. The sequence has the rule aₙ = 6n + 1. Determine whether 103 is a term in the sequence.

9. The recursive rule is:

   a₁ = 8

   aₙ = aₙ₋₁ + 5

   Write an explicit rule for the same sequence.

10. Compare the recursive rule a₁ = 4, aₙ = aₙ₋₁ + 3 with the explicit rule aₙ = 3n + 1. Explain how the two rules describe the same sequence.
11. A geometric sequence begins:

4, 12, 36, 108, ...

Write an explicit rule and use it to find the 8th term.

12. Explain why an explicit rule is generally more efficient than a recursive rule when finding the 500th term of a sequence.