How do I calculate how many push-ups will be done in a year, if I start with 1 a day, the first week, 2/day the second, etc? My son is doing 1 pushup each night before bed, 7 days a week. Then he is adding an additional pushup for the second week (total of 14). Then 3/night the 3rd week, and so on. Can you show me a method of determining how many he will do in a year, if he keeps this up? Thanks.
 A: $365=7 \cdot 52 + 1$. So if we assume that this is NOT a leap year, and your son starts doing pushups on the first day of the year, the number of pushups done = $7(1+2+\dots + 52) + 53$ = $\dfrac{7 \cdot 52 \cdot 53}{2} + 53=9699$. 
If it is a leap year, the number of pushups = $\dfrac{7 \cdot 52 \cdot 53}{2} + 2 \cdot 53=9752$.
A: Your comment indicated that you didn't see how Ayush's answer worked, so here's an explanation.
To make things simple, we'll just look at the number of pushups performed by the end of week 52. In week 1, he'll have done 7, in week 2, at 2 per day, he'll do 14, and so on. The total number of pushups done by the end of week 52 will then be
\begin{align}
P &= 7 + 14 + 21 + \dots + 7\times 50 + 7\times 51 + 7\times 52\\
&= 7 + 14 + 21 + \dots + 350 + 357 + 364
\end{align}
Now add $P$ to itself, written in reverse:
\begin{array}{ccccccccccccc}
P &= 7 &+ &14 &+ &21 &+ &\dots &+ &350 &+ &357 &+ &364\\
P &= 364 &+ &357 &+ &350 &+ &\dots &+ &21 &+ &14 &+ &7
\end{array}
The sum of the first two terms is $(7 + 364) = 371$. That's also the sum of the second two terms, $(14+357)$, since we added and subtracted 7 to the two terms. By the same reasoning, the sum of each of the 52 pairs will also be 371, so when we add all 52 of the pairs we'll have
$$
2P = 371 + 371 + 371 + \dots + 371 + 371 + 371 = 52\times 371
$$
and so $P = (52 \times 371)/2 = 9646$.
This holds in general, too, if you're adding a bunch of terms where each of the adjacent pairs have the same difference (7, in this example, known as an arithmetic sequence): the sum will be
$$
\frac{(\text{number of terms})\times(\text{first term} + \text{last term})}{2}
$$
