Sequences Defined by an Explicit Formula

How To

Given the first few terms of a sequence, find an explicit formula for the sequence.

1. Look for a pattern among the terms.
   
2. If the terms are fractions, look for a separate pattern among the numerators and denominators.
   
3. Look for a pattern among the signs of the terms.
   
4. Write a formula for $a_n$ in terms of $n$. Test your formula for $n=1,n=2$, and $n=3$.
   

Example 4

Writing an Explicit Formula for the $nth$ Term of a Sequence

Write an explicit formula for the $nth$ term of each sequence.

ⓐ $\{−\frac{2}{11},\frac{3}{13},−\frac{4}{15},\frac{5}{17},−\frac{6}{19},…\}$

ⓑ $\{−\frac{2}{25},−\frac{2}{125},−\frac{2}{625},−\frac{2}{3,125},−\frac{2}{15,625},…\}$

ⓒ $\{e^4,e^5,e^6,e^7,e^8,…\}$

Solution

Look for the pattern in each sequence.

ⓐ 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 $2n+9$.

$a_n=\frac{(−1)^n(n+1)}{2n+9}$

ⓑ The terms are all negative.

$\begin{array}{lll} \{ -\frac{2}{25},-\frac{2}{125},-\frac{2}{625},-\frac{2}{3,125},-\frac{2}{15,125},...\} & \text{ the numerator is } 2 \\ \{-\frac{2}{5^2},-\frac{2}{5^3},-\frac{2}{5^4},-\frac{2}{5^6},-\frac{2}{5^7},...-\frac{2}{5^n},\} & \text {the denominators are increasing power of } 5 \end{array}$

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

$a_n=−\frac{2}{5^{n+1}}$

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

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

Try It #4

Write an explicit formula for the $nth$ term of the sequence.

$\{9,−81,729,−6,561,59,049,…\}$

Try It #5

Write an explicit formula for the $nth$ term of the sequence.

$\{−\frac{3}{4},−\frac{9}{8},−\frac{27}{12},−\frac{81}{16},−\frac{243}{20},...\}$

Try It #6

Write an explicit formula for the $nth$ term of the sequence.

$\{\frac{1}{e^2}, \frac{1}{e}, 1, e, e^2,...\}$