Formula for the nth term of a sequence

I need to find a formula for the nth term in the sequence

$$\frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \frac{5}{9}, \frac{6}{11}$$

I have tried the usual approach to find a formula. I first assumed that it was a geometric sequence, but it does not have a common ratio. It is also not an arithmetic sequence. I am at a complete loss on how to find a formula for this.

• Looks like the numerator is incrementing like $n \mapsto n+1$ but the denominator is using $2n + 1 \mapsto 2n + 3$, i.e. walking through odd numbers. – TrostAft Dec 21 '18 at 21:12

$$T_n=\frac{n+1}{2n+1}$$ should be able to work out from the pattern for $$n\in\mathbb{N}$$

HINT:

Numerator are integers beginning at 2. Denominator are odd numbers beginning at 3

1st term numerator--->2

2nd term numerator--->3

3rd term numerator--->4

Can you guess what would be the $$n^{th}$$ term numerator?

1st term denominator is 3=2*1+1

2nd term denominator is 5=2*2+1

3rd term denominator is 7=2*3+1

Can you guess what would be the $$n^{th}$$ term denominator?

• How would I use this information to create a formula? – Elijah Dec 21 '18 at 21:16
• See the edited answer – Tito Eliatron Dec 21 '18 at 21:20