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Explicit Formula of a Sequence — Definition & Examples

Explicit Formula of a Sequence

A formula that allows direct computation of any term for a sequence a1, a2, a3, . . . , an, . . . .

 

Table showing explicit formula a_n = 4n − 7, with n values 1,2,3,4,50,1000 yielding a_n values −3,1,5,9,193,3993.

 

 

See also

Recursive formula

Key Formula

an=f(n)a_n = f(n)
Where:
  • ana_n = The value of the nth term in the sequence
  • nn = The position (index) of the term, usually starting at n = 1
  • f(n)f(n) = A function of n that directly produces the term value

Worked Example

Problem: Find the 20th term of the arithmetic sequence 5, 8, 11, 14, ... using an explicit formula.
Step 1: Identify the first term and the common difference. The first term is a1=5a_1 = 5 and the common difference is d=8−5=3d = 8 - 5 = 3.
a1=5,d=3a_1 = 5, \quad d = 3
Step 2: Write the explicit formula for an arithmetic sequence.
an=a1+(n−1)da_n = a_1 + (n - 1)d
Step 3: Substitute the known values into the formula.
an=5+(n−1)(3)=5+3n−3=3n+2a_n = 5 + (n - 1)(3) = 5 + 3n - 3 = 3n + 2
Step 4: Plug in n=20n = 20 to find the 20th term.
a20=3(20)+2=60+2=62a_{20} = 3(20) + 2 = 60 + 2 = 62
Answer: The 20th term of the sequence is 62.

Another Example

This example uses a geometric sequence instead of an arithmetic one, showing that explicit formulas take different forms depending on the type of sequence. Here the formula involves an exponent rather than a linear expression.

Problem: Find the 8th term of the geometric sequence 3, 6, 12, 24, ... using an explicit formula.
Step 1: Identify the first term and the common ratio. The first term is a1=3a_1 = 3 and the common ratio is r=6/3=2r = 6 / 3 = 2.
a1=3,r=2a_1 = 3, \quad r = 2
Step 2: Write the explicit formula for a geometric sequence.
an=a1⋅r n−1a_n = a_1 \cdot r^{\,n-1}
Step 3: Substitute the known values into the formula.
an=3⋅2 n−1a_n = 3 \cdot 2^{\,n-1}
Step 4: Plug in n=8n = 8 to find the 8th term.
a8=3⋅2 7=3⋅128=384a_8 = 3 \cdot 2^{\,7} = 3 \cdot 128 = 384
Answer: The 8th term of the sequence is 384.

Frequently Asked Questions

What is the difference between an explicit formula and a recursive formula?
An explicit formula computes any term directly from its position number nn, so you can jump straight to, say, the 500th term. A recursive formula defines each term based on one or more previous terms (e.g., an=an−1+3a_n = a_{n-1} + 3), which means you must calculate every term before the one you want. Both describe the same sequence, but the explicit formula is generally faster for finding terms far along in the sequence.
How do you find the explicit formula of a sequence?
First, determine what type of sequence you have. If the differences between consecutive terms are constant, it is arithmetic and the explicit formula is an=a1+(n−1)da_n = a_1 + (n-1)d. If the ratios between consecutive terms are constant, it is geometric and the formula is an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}. For other sequences, look for a pattern relating the term value to nn, such as an=n2a_n = n^2 or an=1na_n = \frac{1}{n}.
Can every sequence be written with an explicit formula?
Not always. Many common sequences (arithmetic, geometric, quadratic, etc.) have clean explicit formulas. However, some sequences defined by complex recursive rules or real-world processes may not have a simple closed-form explicit formula. In those cases, a recursive definition or a table of values may be the most practical way to describe the sequence.

Explicit Formula vs. Recursive Formula

Explicit FormulaRecursive Formula
DefinitionGives the nth term directly as a function of nDefines each term using one or more previous terms
Example (arithmetic)an=5+3(n−1)a_n = 5 + 3(n-1)a1=5,  an=an−1+3a_1 = 5,\; a_n = a_{n-1} + 3
Finding the 100th termSubstitute n = 100 directly — one calculationMust compute all 99 preceding terms first
Ease of writingMay require more work to deriveOften easier to write from a pattern
Best used whenYou need a specific term far into the sequenceYou only need successive terms or the pattern is naturally recursive

Why It Matters

Explicit formulas appear throughout algebra, precalculus, and standardized tests whenever you need to find a specific term of a sequence efficiently. They are essential for modeling real-world situations like compound interest (an=P(1+r)na_n = P(1+r)^n), population growth, and payment schedules, where you need the value at a particular time without computing every intermediate step. Mastering explicit formulas also lays the groundwork for understanding functions, series, and mathematical modeling in higher courses.

Common Mistakes

Mistake: Using nn instead of n−1n - 1 in the exponent or multiplier. For example, writing an=3⋅2na_n = 3 \cdot 2^n instead of an=3⋅2n−1a_n = 3 \cdot 2^{n-1} for the geometric sequence 3, 6, 12, ...
Correction: Always check your formula against the first term. When n=1n = 1, the formula should return a1a_1. In a geometric sequence starting at a1a_1, the exponent is n−1n - 1 because the first term has zero multiplications by rr.
Mistake: Confusing the index nn with the term value ana_n. Students sometimes substitute the term value where nn belongs.
Correction: Remember that nn is the position number (1st, 2nd, 3rd, ...) and ana_n is the output value at that position. When asked 'what position has value 62?', set an=62a_n = 62 and solve for nn, not the other way around.

Related Terms

  • Recursive Formula — Defines terms using previous terms, contrasts with explicit
  • Sequence — The ordered list of numbers described by the formula
  • Arithmetic Sequence — Sequence with constant difference; explicit form is linear
  • Geometric Sequence — Sequence with constant ratio; explicit form is exponential
  • Formula — General term for a mathematical rule or equation
  • Term — Each individual value in a sequence
  • Common Difference — The constant d used in arithmetic explicit formulas
  • Common Ratio — The constant r used in geometric explicit formulas