Hi,

Welcome to BetterExplained! My goal is to help you understand
topics with intuition, not memorization. This is the first in a
series of weekly emails to help you cement your intuition for math.

Let's start with a quick example: How can we add the numbers 1 to
100?

It's a tricky question. At some point, you might have seen the
formula:

total = n * (n + 1) / 2

For example, to add 1 + 2 + 3 ... + 99 + 100, we'd plug in n = 100
and get

100 * 101 / 2 = 5050

Ok, great. We can memorize a one-off formula, but have we
internalized how it works? Does it make us think differently about
the world?

Probably not. A lot of lessons say "Oh, no problem. Here's how you
can figure it out. Pair off numbers at each end: 1 with 100, 2 with
99, 3 with 98, and so on. There are 50 pairs, each is 101, so we
have 50 * 101 = 5050. That's how the formula works."

All set, right? Nope! It still doesn't sit right!

What if we have an odd number of items, like 1 to 99. Does it still
work? Little confusions like this keep us with a mechanical
understanding, not an intuitive one.

I know we can get a solid intuition for this concept. Let's dig
deeper.

Step 1) Simplify The Problem

Why are we adding 1 to 100? It's unmanageable. Let's use 1 to 10.
If we can figure out that simpler case, we can extend to 100.

Step 2) Extract the Intuition

There is something to pairing the numbers to add. But, pairing the
ends makes it difficult to imagine what happens in the middle, and
for even/odd numbers.

How about this setup?

1  2  3  4  5  6  7  8  9  10
10 9  8  7  6  5  4  3  2  1

Oh! If we make two rows -- one forward and one backwards -- we can
easily see the pairs. There's 10 pairs (1 to 10, counted along the
top), and the sum of each is 11.

The total is 10 * 11 = 110, but we only need half, so we get 110 /
2 = 55.

From here, we can internalize the formula as:

Intuition: Count the pairs, multiply by the size of each pair, and
then take half

total = n * (n + 1) / 2

Step 3) Explore Variations

A solid intuition means understanding the result from different
angles.

Geometry Variation

Imagine stacking bricks in a pile, like this:

x
x x
x x x
x x x x
x x x x x

How many are there? 1 + 2 + 3 + 4 + 5. Ok, fine. But is there a
shortcut to counting? Well, imagine completing the wall with 5 more
rows!

x o o o o o
x x o o o o
x x x o o o
x x x x o o
x x x x x o

The wall is now a rectangle: a height of 5, and width of 6. There's
a total of 30 bricks, but we only want half (the x's), so we get 30
/ 2 = 15.

Cool! We can visualize stacking up the bricks, and it might sit
better with us.

Intuition: We have a wall n bricks tall, n + 1 bricks wide, and we
want half

total = n  * (n + 1) / 2

Statistics Variation

What if we don't like visualizing things? No problem. We have a
pattern of items from 1 to n, and the change by the same amount
each time (1, 2, 3...).

The average element in this sequence is

 (first + last) / 2 = (1 + n) / 2

And since we have n elements, we can think of the total like this:

Intuition: Size of average item * number of items

total = (1 + n)/2  * n

Neat!

Each approach expresses the formula differently, such as n *
(n+1)/2 vs (1 + n)/2 * n. That's fine: it shows different
strategies for thinking.

Now, with a solid understanding, we can even think about variations
on the question:

 * How do we add the even numbers up to 10: 2 + 4 + 6 + 8 + 10?
 * How about the odd numbers up to 10: 1 + 3 + 5 + 7 + 9?
 * Or maybe a partial set, like 5 + 6 + 7 + 8 + 9 + 10?

Having an intuition doesn't mean you can spit out the answer in 3
seconds, it's about being comfortable with your approach as you
work through the problem. "Want just the evens less than 10? Ok.
Maybe I'll use the statistics approach: what's the average element
here? How many do we have?".

The formula becomes an expression of the deeper understanding you
have, not a fact floating off in space. I hope this approach works
for you!

Happy math,

-Kalid

PS. Have another idea on how to add 1 to 100? If you know calculus,
you might think of the integral of x.