Sequences and Invariants

A sequence that adds the same amount each time is arithmetic. Seven, eleven, fifteen, nineteen adds four each time, so the common difference is four. To reach the nth term you start at seven and make n minus one jumps of four. That simplifies to four n plus three, so the fiftieth term is two hundred and three. A sequence that multiplies by the same amount each time is geometric. Three, six, twelve, twenty-four doubles each time, so the nth term is three times two to the power n minus one. The tenth term is three times five hundred and twelve, which is one thousand five hundred and thirty-six.

Here is the trick the young Gauss is said to have used. Write one plus two plus all the way to one hundred, then write the same sum backwards underneath. Each column, one and one hundred, two and ninety-nine, and so on, adds to one hundred and one. There are one hundred columns, but that counts the sum twice, so the sum is one hundred times one hundred and one, halved: five thousand and fifty. The same works for any arithmetic series. Count the terms, multiply by the first plus the last, and halve. Seven plus eleven up to two hundred and three has fifty terms, so the sum is fifty times two hundred and ten, halved, which is five thousand two hundred and fifty.

When a pattern repeats, you never need to write it all out. A B C D E repeats in blocks of five, so divide the position by five and look at the remainder. One hundred divided by five is twenty remainder zero. Remainder zero means the hundredth letter finishes a block, so it is E, the last letter. Units digits repeat too. The powers of three end in three, nine, seven, one, then three again, a cycle of four. For three to the power two thousand and twenty-six, divide by four: remainder two, so the units digit is the second in the cycle, nine.

In a growing figure, look at what each new stage adds. A row of matchstick squares starts with four sticks, and each new square shares a stick with its neighbour, so it adds only three. That is an arithmetic sequence: four, seven, ten, and n squares take three n plus one sticks. Some sequences are built from their own earlier terms instead. In the Fibonacci sequence each term is the sum of the two before it: one, one, two, three, five, eight, thirteen. If you know any two neighbouring terms, you can run the rule forwards by adding, or backwards by subtracting.

Some questions ask whether something can ever happen. Cut two opposite corners off a chessboard, leaving sixty-two squares. Can thirty-one dominoes cover them? Trying for an hour proves nothing. Instead, look at the colours. Opposite corners are the same colour, so you are left with thirty dark squares and thirty-two light ones. Every domino, wherever you put it, covers one dark and one light square. Thirty-one dominoes would need thirty-one dark squares, and there are only thirty. So the answer is no, for certain. The colour count is an invariant: a fact that no move can change. Totals and odd-or-even are the other common invariants. Quiz and exercises now.