Sum of arithmetic sequence plus m (Sn + m)

Question

Let $a_n=n$, $a_m=m$. $n\ne m.$ Find $S_{n+m}$.

Should I implement the $S_{n}$ formula and then add m?

[1]: Source :https://i.stack.imgur.com/AZK0Z.png

Answer 1

Think of $S_n$ as a function $S(n)$ that, given a natural number $n$, computes the sum of all natural numbers from 1 to $n$, i.e. $S(n) = 1 + 2 + ... + n$. Note that this is an arithmetic series with $a_1=1, d=1$, so $$S_n = S(n) = \frac{n(2 + (n-1))}{2} = \frac{n(n+1)}{2}, \quad \text{(you can also prove it by induction)}$$ Now, you are asked to find $S_{n+m}$. If you think of $S_n$ as a function that takes integers as input values, then what you need to do is to evaluate $S(n)$ at the point $n+m$: $$S_{n+m} = S(n+m) = \frac{(n+m)(n+m+1)}{2}.$$