Find term n of an arithmetic sequence using the rule

A sequence is an indexed list

Its terms are ordered by an integer index. A finite displayed prefix does not uniquely determine all later terms; a rule or a stated sequence type is needed. Arithmetic means the same difference is added at every step. The difference may be positive, negative or zero

Count steps from the starting index

If the start is u₀ = a and the common difference is d, reaching index n takes n additions. Thus uₙ = a + nd. If instead the first term is called t₁ = a, reaching index n takes n − 1 additions. These conventions describe the same idea but different labels

An arithmetic sequence has u₀ = 7 and d = −3. Find u₅

The first term is t₁ = 7 and d = −3. Which expression gives t₅?

Recover a starting value from a later term

If uₖ and d are known with index origin zero, reverse k additions: a = uₖ − kd. For u₄ = 19 and d = 3, a = 7. Then u₈ = 31. Check the recovered rule against the supplied term before using it elsewhere

Derive the term formula, ungraded

Prove uₙ = a + nd for the recurrence u₀ = a, uₙ₊₁ = uₙ + d, for all nonnegative integers n. State a base case and an arbitrary-index induction step

Model: an induction on the index

At n = 0 the formula gives a, matching u₀. Suppose uₖ = a + kd for an arbitrary k ≥ 0. The recurrence gives uₖ₊₁ = uₖ + d = a + kd + d = a + (k + 1)d. This proves the next case from the hypothesis, so induction establishes the formula for every permitted index. No sign restriction on a or d is needed

Use the gap between two term indices

Two known terms can determine both the first term and the common difference of a stated arithmetic sequence. Between indices p and q there are q − p updates, not q − p + 1. Subtracting the term equations removes the unknown start; divide the change in value by the number of updates

An arithmetic sequence has t₄ = 18 and t₉ = −7. Its first term is t₁ = a and its common difference is d. Find the ordered pair (a, d)

Model: subtract, recover and check

With first index 1, the two observations give a + 3d = 18 and a + 8d = −7. Subtract the first equation from the second to find d, then substitute to recover a. The rule must reproduce both observations, not just one

An arithmetic sequence instead has u₄ = 18 and u₉ = −7, with its starting value labelled u₀. What is u₀?

Explain uniqueness and the index origin, ungraded

For a stated arithmetic sequence, explain why two observations at different indices determine one common difference and one starting value. For values 18 at index 4 and −7 at index 9, explain why the first term is 33 when counting from index 1, while the starting term is 38 when counting from index 0. What changes if both observations use the same index, or if no arithmetic condition is supplied?

Model: the assumption and index matter

The arithmetic sequence has value 18 at index 4 and −7 at index 9; five updates change its value by −25, so d = −5. Different indices give a nonzero divisor q − p, so subtraction determines exactly one d. Either observation then determines the chosen start. Here t₁ = 33 but extending the same rule to index zero gives 38: they are different indexed terms. Two equal observations at the same index leave d undetermined; different values at the same index are inconsistent. Without the arithmetic assumption, other rules can agree at both observed indices and differ elsewhere