Question

Determine the first four terms of the sequence in which the nth term is an=(n+3)!/(n+4)!

Answer 1

In this case, we'll have to carry out several steps to find the solution.

Step 01:

Data:

an=(n+3)!/(n+4)!

Step 02:

nth term sequence:

1. n = 1

`a_1=((1+3)!)/((1+4)!)=(4!)/(5!)=(4\cdot3\cdot2\cdot1)/(5\cdot4\cdot3\cdot2\cdot1)=(1)/(5)`

2. n = 2

`a_2=((2+3)!)/((2+4)!)=(5!)/(6!)=(5\cdot4\cdot3\cdot2\cdot1)/(6\cdot5\cdot4\cdot3\cdot2\cdot1)=(1)/(6)`

3. n = 3

`a_3=((3+3)!)/((3+4)!)=(6!)/(7!)=(6\cdot5\cdot4\cdot3\cdot2\cdot1)/(7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1)=(1)/(7)`

4. n = 4

`a_4=((4+3)!)/((4+4)!)=(7!)/(8!)=(7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1)/(8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1)=(1)/(8)`

The answer is:

`(1)/(5),(1)/(6),(1)/(7),(1)/(8)`