Equal steps vs doubling

A clerk adds ten coins to a jar every Friday. After ten weeks the jar holds one hundred coins. Next Friday will add ten more. That is a linear climb: the same step, over and over.

Now change the rule. Start with one coin. Each Friday, double whatever is in the jar. After ten Fridays the jar holds 512 coins, not 100. The eleventh Friday adds 512 more. That is doubling. People call a climb like this exponential when the next amount is the current amount multiplied, not added. The interesting number sits in the exponent.

At the start, doubling can look smaller than the equal steps. One, two, four, eight — the ten-coin Friday is still ahead. Early doubling sits under a straight line. The line looks like it is winning. It is not, for long.

Once you can see the two rules, a quiet week is not proof that nothing is happening. It may be the part of a doubling that still looks small.

The tempting mistake is to treat every climb as equal steps, because walking, wages, and school years feel that way.

That is what lets you ask what a board of doublings actually holds.