Write an expression for the nth term in the sequence 3, 5, 7, 9, 11, … and find the value of the 100th term.

Hint:

The terms are all odd numbers.  - First look similarities and differences between the terms.
Each term is greater than the one before it.
The difference between one term and the next one is 2.
Assume that each term is going to be 2 more than the one before it. (They will all continue to be odd) - Generalize the observations.
The next terms will be 13, 15, and 17.
Terms 1 is 3.
Terms 2 is 5.
Terms 3 is 7.
Term 4 is 9.   - Write a conjecture. Since the problem asks for the nth term, we want an algebraic expression that connects the position of the term in the sequence to the term's value.
As the term position goes up by 1, the term value goes up by 2.
Try multiplying the term position by2:
Term 1 gives 2.
Term 2 gives 4.
Term 3 gives 6.
Each of these is too low by 1, so add 1:
Term 4 is 2 × 4 + 1 = 9. That worked!
The nth term has  value 2n + 1.   - Make the prediction requested by the problem. (This step may not always be needed.)
The 100th term has a value of 2(100) + 1, or 201.
The nth term is 2n + 1, and the 100th term is 201.