9.1 Sequences and their notations  (Page 3/15)

Substitute $n=1,$ $n=2,$ and so on in the formula.

The first five terms are $\left\{-\frac{1}{2},\frac{4}{3},-\frac{9}{4},\frac{16}{5},-\frac{25}{6}\right\}.$

Substitute $n=1,n=2,$ and so on in the appropriate formula. Use $n^2$ when $n$ is not a multiple of 3. Use $\frac{n}{3}$ when $n$ is a multiple of 3.

The first six terms are $\left\{1,4,1,16,25,2\right\}.$

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 $n+3.$

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