Key takeaway: An explicit formula gives the nth term of a sequence directly as an expression in n, so any term can be found without computing the terms before it.
Grade 9 – Algebra 1F-BF.A.2Functions and Graphs
Definition of Explicit Formula
An explicit formula is a rule that takes the position number \(n\) as its input and returns the term at that position. For the sequence 11, 7, 3, \(-1\), the rule \(a_n = 15 – 4n\) works: put \(n = 1\) and you get 11, put \(n = 2\) and you get 7.
The big advantage is speed. To find the 40th term, you evaluate the formula once instead of listing 39 terms first. You can also run it backward, solving for \(n\) to learn where a given value appears. Arithmetic sequences have linear explicit formulas, and geometric sequences have exponential ones.
Formula and notation
Write the formula with the position as the variable:
\[a_n = 15 – 4n\]
It is function notation whose inputs are the counting numbers 1, 2, 3 and so on. In general, arithmetic sequences use \(a_n = a_1 + (n-1)d\) and geometric ones use \(a_n = a_1 r^{\,n-1}\).
Key facts
- The input of an explicit formula is the term number n, a positive integer.
- Evaluating it once gives any term; no earlier terms are needed.
- Setting the formula equal to a value and solving tells which term holds it.
- Test a proposed formula by checking that n = 1 and n = 2 give the listed terms.
Explicit Formula example
The sequence 11, 7, 3, \(-1\) and so on has the explicit formula \(a_n = 15 – 4n\). Find the 40th term, and find which term equals \(-45\).
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40th term: \(a_{40} = 15 – 4(40) = 15 – 160 = -145\).
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Set the rule equal to \(-45\): \(15 – 4n = -45\).
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Solve: \(-4n = -60\), so \(n = 15\).
The 40th term is \(-145\), and \(-45\) is the 15th term.
Common mistake
Watch out
Seeing that the sequence starts at 11 and drops by 4, Ines writes \(a_n = 11 – 4n\).
Why it is wrong
Put \(n = 1\) into her rule: \(11 – 4 = 7\), but the first term is 11. The constant in this style of formula is not the first term; it is the value one step before it, \(11 + 4 = 15\). Writing \(a_n = 11 – 4(n – 1)\) also works and simplifies to \(15 – 4n\).
Related terms and lessons
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