Exponential and combined sequence rules

When each term is obtained by multiplying the previous term by a fixed ratio r, an nth-term rule is \(T_n=ar^{n-1}\), where a is the first term. This follows from taking n−1 multiplication steps.

Example

For \(3,6,12,24,\ldots\), \(T_n=3(2^{n-1})\). For \(3,6,11,20,37,\ldots\), compare with \(2^n\): the differences are \(1,2,3,4,5\), so \(T_n=2^n+n\).

Alternating signs can be represented by \((-1)^n\) or \((-1)^{n+1}\). The rule \(T_n=(-1)^{n+1}n\) gives \(1,-2,3,-4,\ldots\). Check the first term to choose the correct sign.

For sequences beginning at \(n=1\), a proposed term requires a positive integer position. Use nth-term rules here; a total of terms is a separate question.