Find a formula for the $n$ th term of the sequence where $a_{n}$ is calculated directly from $n$ .
$\frac{7}{1}, \frac{11}{2}, \frac{15}{6}, \frac{19}{24}, \frac{23}{120}$

Answer

Expert–verified

Hide Steps

Answer

Final Answer: The formula for the nth term of the sequence is $\frac{3 + 4 n}{n!}$

Steps

Step 1 ：Observe the given sequence: $\frac{7}{1}, \frac{11}{2}, \frac{15}{6}, \frac{19}{24}, \frac{23}{120}$

Step 2 ：Notice that the numerator of each term is increasing by 4 and the denominator seems to be the factorial of the term number.

Step 3 ：Attempt to formulate a general formula for the nth term of the sequence.

Step 4 ：Based on the pattern, the formula for the nth term of the sequence is $\frac{3 + 4 n}{n!}$

Step 5 ：Final Answer: The formula for the nth term of the sequence is $\frac{3 + 4 n}{n!}$