Explicit Formula

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Explicit Formula: meaning, method and solved example (Algebra 1) – Maths PDF

In short: An explicit formula is a rule that turns a position number n straight into the value at that position, with no need to compute earlier entries.

Algebra 1Functions and GraphsF-BF.A.2

What is Explicit Formula?

Skipping the walk down the list is the whole point of an explicit formula: insert the position, receive the value. For 7, 11, 15, 19, the rule \(a_n = 4n + 3\) reports the 20th entry as \(4(20) + 3 = 83\) immediately.

Arithmetic lists follow \(a_n = a_1 + (n – 1)d\), and geometric lists follow \(a_n = a_1 \cdot r^{n-1}\). Since the rule is a function of n, you can also run it backward to learn which position holds a chosen value.

How to spot it

  • A question asks for a term far down the list, such as the 40th or the 200th.
  • An explicit formula appears as a rule with n alone on the right, with no earlier term in it.
  • You are asked which term of a sequence equals a particular number.
  • A pattern is described by its position, like the nth row of seats.

The method

  1. Decide whether the pattern adds a constant (arithmetic) or multiplies by one (geometric).
  2. Find \(a_1\) and either the common difference d or the common ratio r to build the explicit formula.
  3. Substitute into \(a_1 + (n – 1)d\) or \(a_1 r^{n-1}\) and simplify to a rule in n.
  4. Test the rule on the first two terms, then evaluate it at the position you need.
naₙ12345711151923aₙ = 4n + 3a₂₀ = 4(20) + 3 = 83
No need to list 19 terms first: n = 20 goes straight into the rule.

Explicit Formula: solved example

Problem

A list begins \(9, 14, 19, 24, \ldots\). Give a rule for the nth entry, compute the 40th entry, and locate the entry equal to 134.

Solution

The common difference is \(d = 5\) and \(a_1 = 9\), so \(a_n = 9 + (n – 1)(5) = 5n + 4\).

The 40th term is \(5(40) + 4 = \mathbf{204}\). For 134: \(5n + 4 = 134\), so \(5n = 130\) and \(n = 26\). The formula is \(a_n = 5n + 4\), \(a_{40} = 204\), and 134 is term 26.

Try it

Question 1

Find a rule in n for \(3, 6, 12, \ldots\), then use it for the 8th entry.

Show the answer

\(a_n = 3 \cdot 2^{n-1}\), so \(a_8 = 3 \cdot 128 = 384\).

Question 2

If \(a_n = n^2 – 1\), what is \(a_7\)?

Show the answer

\(7^2 – 1 = 48\).

Explicit Formula vs. Recursive Formula

This kind of rule maps a position straight to a value, like a function of n. A recursive rule instead says how to get from one term to the next and needs a start. If you need term 500, the explicit version answers in one calculation, while the recursive version needs 499 earlier steps.

Recursive Formula

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