Gauss and the lazy sum
Tonight's idea
The boy was Carl Friedrich Gauss, and the story may be polished, but the trick is real. Instead of adding in order, pair the ends: \(1 + 100 = 101\), \(2 + 99 = 101\), \(3 + 98 = 101\), and so on. There are 50 pairs, each worth 101, so the total is \(50 \times 101 = 5050\).
In general, adding \(1 + 2 + \cdots + n\) gives \(\dfrac{n(n+1)}{2}\). You can see why with a picture instead of pairs: draw the sum as a staircase of dots, 1 dot in the first row, 2 in the second, up to \(n\). Two staircases fit together into an \(n\) by \((n+1)\) rectangle, so one staircase is half of that.
These totals, 1, 3, 6, 10, 15, 21, …, are the triangular numbers, because the staircase is a triangle of dots. They turn up everywhere: the number of handshakes in a room of \(n+1\) people, the number of balls in a rack of pool, a diagonal of Pascal's triangle.
Watch one: add 11 + 12 + … + 30
Pairing works, but subtraction is lazier still. The sum from 1 to 30 is \(\dfrac{30 \times 31}{2} = 465\). The sum from 1 to 10 is \(\dfrac{10 \times 11}{2} = 55\). The sum from 11 to 30 is everything in the first total that is not in the second: \(465 - 55 = 410\).
Check by pairing: \(11 + 30 = 41\), \(12 + 29 = 41\), … There are 20 numbers, so 10 pairs: \(10 \times 41 = 410\). Same answer.
Your turn
- Use the formula to find \(1 + 2 + \cdots + 10\), then \(1 + 2 + \cdots + 1000\). Check the first one by hand.
- Add the even numbers \(2 + 4 + 6 + \cdots + 100\). (Hint: factor out a 2, and you are left with a sum you already know how to do.)
- Add \(21 + 22 + \cdots + 60\) using the subtraction method.
- Add the odd numbers \(1 + 3 + 5 + \cdots + 19\). Then add \(1 + 3\), \(1 + 3 + 5\), \(1 + 3 + 5 + 7\). What kind of numbers do you keep getting? How many odd numbers were in the first sum, and how does that predict its total?
- Everyone at a party of 100 people shakes hands with everyone else exactly once. How many handshakes? (The first person shakes 99 hands, the next shakes 98 new ones, …)
- The picture in Tonight's idea shows \(n = 3\). Draw the same thing for \(n = 4\): a staircase with rows of 1, 2, 3, 4 dots, and a second copy upside down beside it. What rectangle do you get, and how does it give \(\dfrac{4 \times 5}{2}\)?
Stretch: why odd numbers make squares
Look at the picture. Start with one square. Wrap an L of 3 squares around it, then an L of 5, then 7, then 9. Why is each new L exactly 2 more squares than the last? Explain, in a sentence or two, why \(1 + 3 + 5 + \cdots + (2n-1) = n^2\) for every \(n\).
Then try this one with a calculator and no proof needed: compute \(1^3\), \(1^3 + 2^3\), \(1^3 + 2^3 + 3^3\), \(1^3 + 2^3 + 3^3 + 4^3\). Compare with the triangular numbers 1, 3, 6, 10. What do you notice?
Answers
- \(\frac{10 \times 11}{2} = 55\) and \(\frac{1000 \times 1001}{2} = 500{,}500\).
- \(2 + 4 + \cdots + 100 = 2(1 + 2 + \cdots + 50) = 2 \times 1275 = 2550\).
- \(\frac{60 \times 61}{2} - \frac{20 \times 21}{2} = 1830 - 210 = 1620\).
- \(1 + 3 + \cdots + 19 = 100\). The short sums are 4, 9, 16. They are all perfect squares. There are 10 odd numbers in the first sum, and \(10^2 = 100\): the sum of the first \(n\) odd numbers is \(n^2\).
- \(99 + 98 + \cdots + 1 = \frac{99 \times 100}{2} = 4950\).
- Two staircases of 1, 2, 3, 4 fit together into a 4 by 5 rectangle of 20 dots. One staircase is half, 10 dots, which is \(\frac{4 \times 5}{2}\). The general picture is \(n\) by \(n+1\).
Stretch. Each L wraps around a square that is one unit bigger on each side, so it needs one extra square along the bottom and one extra up the side: 2 more than the previous L. The L around an \((n-1)\) by \((n-1)\) square has \(2n - 1\) cells: \(n\) along the bottom, \(n\) up the side, minus the shared corner. Adding the L turns an \((n-1)^2\) square into an \(n^2\) square, so the odd numbers add up to squares forever. The cubes: 1, 9, 36, 100, which are \(1^2, 3^2, 6^2, 10^2\), the squares of the triangular numbers. So \(1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2\). That one is true for every \(n\); there is a picture proof, for another night.