Repetitions are Sequences

When doing a task like working out, a common pattern is to perform something like 100 reps, then 90 reps, then 80, and so on, until you’ve completely counted down to zero. But this pattern can also be expressed arithmetically.

We say that there are 11 terms in this sequence: 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, and 0. Alternatively, we could count the terms by solving:

\[ 0 = 100 - 10 (n - 1) \]

Now, let S represent the sum of all the terms in our sequence, N represent the number of terms, and \( t_0 \) and \( t_1 \) represent the first and last terms of the sequence.

\[ S_n = \frac{n}{2} \cdot (t_1 + t_n) \]

If we're beginning at 100 and counting all the way down to zero, we plug those values into our equation to get the total sum of 550.

\[ S_n = \frac{11}{2} \cdot (100 + 0) = 550 \]