Arithmetic sequences.

Step 1 of 5

Add the difference

A sequence is a list of numbers in order: $a_1$⁠ is the first term, $a_2$⁠ the second, and so on. In an arithmetic sequence each term is more than the one before by the same number, the difference $d$⁠: $a_{n+1} = a_n + d$⁠. On the board each term is a column of blocks, and the arithmetic sequence is the staircase on the left. Here $a_1 = 1$⁠ and Difference d is $2$⁠, and only the first term is shown. Predict the first five terms.

Step 2 of 5

The general term

To reach the $n$⁠th term you need not go through every term before it. From the first term to the $n$⁠th there are $n - 1$⁠ steps, and each step adds $d$⁠:

$a_n = a_1 + (n - 1)d$$a_n =$$a_1 + (n - 1)d$⁠

This formula is a linear function of $n$⁠, as in the linear functions Unit: the difference $d$⁠ is the slope, which is why the column tops climb in a straight line. Terms n is locked at $8$⁠; find the difference that gives a particular eighth term.

Step 3 of 5

Terms far away

With the formula you can jump straight to a far-off term, even though the board shows only ten. In the sequence of odd numbers, $a_1 = 1$⁠ and $d = 2$⁠, the tenth term is $1 + 9 \cdot 2 = 19$⁠, the number above the last column. Work out a much later term the same way.

Step 4 of 5

Adding in pairs

What is $1 + 2 + \dots + 100$⁠? A famous story says Carl Friedrich Gauss solved it as a boy in minutes: he added the first and the last, $1 + 100 = 101$⁠, the second and the second to last, $2 + 99 = 101$⁠, and so on. There are $50$⁠ such pairs, so the sum is $50 \cdot 101 = 5050$⁠.

The same idea works for any arithmetic sequence, because every “first plus last” pair is equal. Write $S_n$⁠ for the sum of the first $n$⁠ terms:

$S_n = \frac{n(a_1 + a_n)}{2}$$S_n =$$\frac{n(a_1 + a_n)}{2}$⁠

Here the sequence is $1, 2, \dots, 10$⁠. Work out $S_{10}$⁠, then move Add the terms to the running totals: each column lifts onto itself every term before it and shows the running total.

Step 5 of 5

Odd numbers and squares

The board shows the running totals of the odd numbers: $1$⁠, $1 + 3 = 4$⁠, $1 + 3 + 5 = 9$⁠. The totals do not climb in a straight line but in a curve: they are squares. Move Terms n to find how many odd numbers add up to $49$⁠.