Sequence Sleuth

Sequence Sleuth 🔍

A **sequence** is a list of numbers following a rule. Your job is to uncover that rule — then you can continue the list, or jump straight to the 100th term without writing them all out. Two clues crack most cases: the **term-to-term** rule and the **nth-term** formula.

Term-to-term rules 🔢

The **term-to-term** rule tells you how to get from one term to the next: • **Arithmetic** — **add** a constant each time. 3, 5, 7, 9 ... (add 2). • **Geometric** — **multiply** by a constant each time. 2, 6, 18, 54 ... (× 3). Spot which is happening and you can extend the sequence as far as you like.

Find the next term

An interactive activity.

What is the rule?

Look at the sequence 2, 6, 18, 54, ... What is the term-to-term rule?

• Multiply by 3
• Add 4
• Add the two previous terms
• Multiply by 2

The nth-term formula 🎯

The **nth term** lets you find *any* term from its position n. For a linear (arithmetic) sequence: **nth term = (common difference) × n + (first term − common difference).** For 3, 5, 7, 9: the difference is 2, and 3 − 2 = 1, so the nth term is **2n + 1**. Check: n = 1 gives 2(1) + 1 = 3. ✓

Order the method

An interactive activity.

The coefficient of n

An interactive activity.

Jump to the 10th term

An interactive activity.

Match each special sequence to its name

The sleuth rules

In an arithmetic sequence you _____ a constant amount each time; in a geometric sequence you _____. The nth term of a linear sequence has the common _____ as the number in front of n.