A sequence is an ordered list. The position n and the term T_n are different: T_1 is the first term. A term-to-term rule finds the next term from the previous one; an nth-term rule finds a term directly from its position. Example For 5,9,13,17,, add 4 each time. Compare with 4n, whose values are 4,8,12,16,. Every term is 1 larger, so T_n=4n+1. T_20=4(20)+1=81. For a constant difference d and first term a, T_n=a+(n-1)d: reaching position n takes n-1 steps. A few terms alone can fit many rules. Use the stated pattern or construction, and check a proposed rule against every given term.

A sequence is an ordered list. The position \(n\) and the term \(T_n\) are different: \(T_1\) is the first term. A term-to-term rule finds the next term from the previous one; an nth-term rule finds a term directly from its position.

For \(5,9,13,17,\ldots\), add \(4\) each time. Compare with \(4n\), whose values are \(4,8,12,16,\ldots\). Every term is \(1\) larger, so \(T_n=4n+1\).

\(T_{20}=4(20)+1=81\). For a constant difference \(d\) and first term \(a\), \(T_n=a+(n-1)d\): reaching position \(n\) takes \(n-1\) steps.

A few terms alone can fit many rules. Use the stated pattern or construction, and check a proposed rule against every given term.

From a sequence to a position rule

From a sequence to a position rule