Writing an Explicit Formula for the nth Term of a Sequence

Look for the pattern in each sequence.

1. ⓐThe terms alternate between positive and negative. We can use ${(-1)}^n$ to make the terms alternate. The numerator can be represented by $n+1.$ The denominator can be represented by $2n+9.$

   $a_n=\frac{{(-1)}^n(n+1)}{2n+9}$

2. ⓑ

   The terms are all negative.

   

   So we know that the fraction is negative, the numerator is 2, and the denominator can be represented by $5^{n+1}.$

   $$a_n=-\frac{2}{5^{n+1}}$$
3. ⓒ

   The terms are powers of $e.$ For $n=1,$ the first term is $e^4$ so the exponent must be $n+3.$

   $$a_n=e^{n+3}$$