63.3 Arithmetic Explicit Rule

The Arithmetic Explicit Rule defines how to calculate terms in a sequence using a direct formula, providing clear steps for each term's value.

Arithmetic Explicit Rule is a formula that calculates any term of an arithmetic sequence directly from its position number, without requiring the calculation of any of the terms that come before it, built from the sequence's first term and its constant common difference.

Arithmetic First-Term Identification

Identifying the Starting Value

The first term of the sequence, the value at position one, is identified directly from the given sequence as the starting point for the explicit rule.

a_{1}

Role of the First Term in the Rule

This value serves as the fixed anchor point from which every other term is calculated, since the explicit rule measures how far any given term has moved away from this specific starting value.

Arithmetic Common-Difference Identification

Identifying the Constant Difference

The common difference, the fixed amount added between any two consecutive terms, is identified using the same difference calculation and confirmation process established during pattern recognition.

d = a_{n} - a_{n - 1}

Role of the Common Difference in the Rule

This value determines the rate at which the sequence grows or shrinks, and it is the quantity that will be scaled according to position when building the full explicit rule.

Sequence Position Offset

Measuring Distance from the First Term

The position offset is the number of steps between the first term and the term being calculated, found by subtracting one from the desired position number.

n - 1

Why the Offset Subtracts One

Because the first term itself requires zero steps of change away from itself, the offset must be one less than the position number, ensuring that position one corresponds to zero applications of the common difference.

Common-Difference Offset Product

Multiplying the Offset by the Difference

The position offset is multiplied by the common difference, producing the total amount of change that has accumulated between the first term and the term at the desired position.

(n - 1) d

Why Multiplication Represents Total Change

Since the same fixed difference is added at every single step from the first term onward, multiplying that fixed value by the number of steps taken is what correctly totals the accumulated change.

Arithmetic Nth-Term Rule Construction

Assembling the Full Explicit Rule

The explicit rule is constructed by adding the total accumulated change to the first term, producing a single formula for any term based on its position number.

a_{n} = a_{1} + (n - 1) d

Why This Construction Is Complete

This formula combines the fixed starting value with the exact amount of accumulated change for any position, meaning no further information beyond the first term and the common difference is needed to describe every term in the sequence.

Arithmetic Term Evaluation

Finding a Term from Its Position

Given a specific position number, the explicit rule is evaluated by substituting that number for n and simplifying the resulting expression.

a_{10} = a_{1} + 9 d

Efficiency Compared to Repeated Addition

This direct evaluation reaches any term in a single calculation, avoiding the need to repeatedly add the common difference one step at a time from the first term up to the desired position.

Arithmetic Term Index Determination

Finding the Position of a Known Term

Given a specific term value, the explicit rule is solved for n instead of for the term itself, determining which position that value occupies within the sequence.

n = \frac{a_{n} - a_{1}}{d} + 1

Reversing the Direction of the Rule

This use of the explicit rule reverses its typical direction, treating the term value as the known quantity and the position number as the unknown quantity to be isolated and solved for.

Explicit Rule Sequence Check

Confirming the Rule Reproduces Known Terms

The constructed explicit rule is checked by substituting a few known position numbers, such as one, two, and three, and confirming that the resulting values match the original given terms of the sequence.

Why This Check Matters

Because the explicit rule depends on both the first term and the common difference being extracted correctly, testing it against terms that are already known confirms that no error was introduced during identification or construction before the rule is used to find any unknown term.