Question

Here are the first four terms in a sequence.

1/2, 4/3, 9/4 and 16/8
Find the nth term of this sequence.

Answer 1

`(n^2)/(n+1)`

Explanation:

Let's break this apart by numerator (N) and denominator (D) and see if we can spot a pattern:

• 1st term: N = 1, D = 2
• 2nd term: N = 4, D = 3
• 3rd term: N = 9, D = 4
• 4th term: N = 16, D = 5

Don't see it? Let's write it another way:

• 1st: N = 1², D = 1 + 1
• 2nd: N = 2², D = 2 + 1
• 3rd: N = 3², D = 3 + 1
• 4th: N = 4², D = 4 + 1

The numerator always seems to be the number of the term squared, while the denominator is the number of the term plus 1. Putting things back in fraction form, we can say that

• nth term:
  `(n^2)/(n+1)`