Q. 30 - Calculus

Question

In Exercises, find a plausible formula for the general term of the given sequence.

$\left\{1,\frac{1}{2},\frac{1}{6},\frac{1}{24},\frac{1}{120},\dots \right\}$

Step-by-Step Solution

Answer

The plausible formula for the general term of the sequence is $a_k=\frac{1}{k!}$where $k\ge 1.$

Step 1. Given information.

The given sequence is $\left\{1,\frac{1}{2},\frac{1}{6},\frac{1}{24},\frac{1}{120},\dots \right\}.$

Step 2. plausible formula for the general term.

write the sequence in a pattern. 

$$\begin{gathered} \left\{1,\frac{1}{2},\frac{1}{6},\frac{1}{24},\frac{1}{120},\dots \right\}=\left\{\frac{1}{1!},\frac{1}{2!},\frac{1}{3!},\frac{1}{4!},\frac{1}{5!},\dots \right\} \\ \left\{1,\frac{1}{2},\frac{1}{6},\frac{1}{24},\frac{1}{120},\dots \right\}={\left\{\frac{1}{k!}\right\}}_{k=1}^{\infty } \end{gathered}$$

for every term, the numerator is $1$and the denominator is $k!$and $k\ge 1.$

So plausible formula for the general term of the sequence is  $a_k=\frac{1}{k!}.$