summation - Formula for sum of first $n$ odd integers

I'm self-studying Spivak's Calculus and I'm currently going through the pages and problems on induction. This is my first encounter with induction and I would like for someone more experienced than me to give me a hint and direction.
The first problem is as follows:

Find a formula for

$$\sum_{i=1}^n(2i-1)=1+3+5+...+(2n-1)$$
And the related following problem:

Find a formula for

$$\sum_{i=1}^n(2i-1)^2=1^2+3^2+5^2+...+(2n-1)^2$$

The given hints are: "What do these expressions have to do with $1+2+3+...+2n$ and $1^2+2^2+3^2+...+(2n)^2$?"

I recognize that the above sums are the sum of all the odd integers from $1$ to $n$ and the sum of all the squares of the odd integers from $1$ to $n$, respectively. My question is, in problems like these does one just do a bunch of trial and error, as I have done for quite a while now, or is there a more clever way to go about it?