Solve a Simple Linear Recurrence

1) Identify the recurrence type

A simple linear recurrence usually gives each term from the previous one, often with a constant multiplier or a constant added term. Your goal is to rewrite it until you can see the pattern and express the term directly.

2) Build the first few terms

Start from the given initial value and compute the next terms step by step. This helps you detect whether the sequence is:

• multiplicative: each term is multiplied by the same number,
• additive: each term increases by the same amount,
• or a combination of both.

3) Look for a closed form

For a multiplication-only recurrence, repeated substitution often gives a power pattern. For an additive recurrence, the terms usually form an arithmetic pattern. If both a multiplier and constant appear, try separating the steady part from the changing part by testing a shifted variable.

4) Simplify carefully

Once you get an expression, simplify it fully. Combine powers, reduce fractions, and write the answer in its cleanest exact form.

5) Check your result

Substitute your formula back into the recurrence and verify it produces the next term. Also test the initial value to make sure the formula matches from the start.