how to solve 100^(1%4)?
Answers
Putting this into our formula
\displaystyle{\text{sum = average * count } = \frac{(1 + n)}{2} \cdot n = \frac{n(n + 1)}{2}}
And voila! We have a fourth way of thinking about our formula.
So Why Is This Useful?
Three reasons:
1) Adding up numbers quickly can be useful for estimation. Notice that the formula expands to this:
\displaystyle{\frac{n(n+1)}{2} = \frac{n^2}{2} + \frac{n}{2} }
Let’s say you want to add the numbers from 1 to 1000: suppose you get 1 additional visitor to your site each day – how many total visitors will you have after 1000 days? Since thousand squared = 1 million, we get million / 2 + 1000/2 = 500,500.
2) This concept of adding numbers 1 to N shows up in other places, like figuring out the probability for the birthday paradox. Having a firm grasp of this formula will help your understanding in many areas.
3) Most importantly, this example shows there are many ways to understand a formula. Maybe you like the pairing method, maybe you prefer the rectangle technique, or maybe there’s another explanation that works for you. Don’t give up when you don’t understand — try to find another explanation that works. Happy math.
By the way, there are more details about the history of this story and the technique Gauss may have used.
Variations
Instead of 1 to n, how about 5 to n?
Start with the regular formula (1 + 2 + 3 + … + n = n * (n + 1) / 2) and subtract off the part you don’t want (1 + 2 + 3 + 4 = 4 * (4 + 1) / 2 = 10).
Sum for 5 + 6 + 7 + 8 + … n = [n * (n + 1) / 2] – 10
And for any starting number a:
Sum from a to n = [n * (n + 1) / 2] – [(a - 1) * a / 2]
We want to get rid of every number from 1 up to a – 1.
How about even numbers, like 2 + 4 + 6 + 8 + … + n?
Just double the regular formula. To add evens from 2 to 50, find 1 + 2 + 3 + 4 … + 25 and double it:
Sum of 2 + 4 + 6 + … + n = 2 * (1 + 2 + 3 + … + n/2) = 2 * n/2 * (n/2 + 1) / 2 = n/2 * (n/2 + 1)
So, to get the evens from 2 to 50 you’d do 25 * (25 + 1) = 650
How about odd numbers, like 1 + 3 + 5 + 7 + … + n?
That’s the same as the even formula, except each number is 1 less than its counterpart (we have 1 instead of 2, 3 instead of 4, and so on). We get the next biggest even number (n + 1) and take off the extra (n + 1)/2 “-1″ items:
Sum of 1 + 3 + 5 + 7 + … + n = [(n + 1)/2 * ((n + 1)/2 + 1)] – [(n + 1) / 2]
To add 1 + 3 + 5 + … 13, get the next biggest even (n + 1 = 14) and do
[14/2 * (14/2 + 1)] – 7 = 7 * 8 – 7 = 56 – 7 = 49
Combinations: evens and offset
Let’s say you want the evens from 50 + 52 + 54 + 56 + … 100. Find all the evens
2 + 4 + 6 + … + 100 = 50 * 51
and subtract off the ones you don’t want
2 + 4 + 6 + … 48 = 24 * 25
So, the sum from 50 + 52 + … 100 = (50 * 51) – (24 * 25) = 1950
Phew! Hope this helps.