sequences

[xc630]

Hello I need help coming up with rules for the folowing two sequences.

1/2, 2/3, 3/4, 4/5...

In this sequence I noticed that the numerators and denominators each increase by one each time. ALso the denominator is the numerator of the next term and the the denominator of that term is 1 greater thean the numerator. I don't know hwo to write a rule though.

2/1, 3/4, 4/9, 5/16...

In this one I see that the numerators increase by 1 and the denominators increase by increasing odd numbers (3,5,7).

Any help would be appreciated.

 

[pka]

1/2, 2/3, 3/4, 4/5...

Note that the numerators are the numbers of the terms.
The denominators are just one more. \(\displaystyle \L
a_n = \frac{n}{{n + 1}}\)

2/1, 3/4, 4/9, 5/16...

Look at the denominators. They are the squares of the number of the term.
The numerators are just one more.
\(\displaystyle \L
b_n = \frac{{n + 1}}{{n^2 }}\).