173. Offspring of a Cow. "Suppose," said my friend the farmer Hodge, "that my cow gives birth to a heifer at the age of two. Suppose also that she will give birth to a heifer every year and that each heifer, upon reaching the age of two, will follow her mother's example and give birth to a heifer annually, and so on. Now, tell me, how many offspring will this cow have in 25 years?

From Hodge's explanation, it was clear that he was counting the time from the birth of the very first cow and that over the entire 25 years, he would not have any of his own beef or veal.[^9]
173. Let's write down the following sequence of numbers, first studied by Leonardo Fibonacci (born in 1175), who almost introduced the familiar Arabic numerals to Europe:

\[
0,1,1,2,3,5,8,13,21,34, \ldots, 46368
\]

Each subsequent number is the sum of the two preceding ones. The sum of all numbers from the first to the given one is 1 less than the number that follows the given one by two. If you double any member of the sequence and add the previous one, you get the member that follows the given one by two. Further, in the first year, the offspring will be 0 calves, in the second 1, in the third 1, in the fourth 2, and so on. This way, the members of this sequence will be obtained. The twenty-fifth member is 46368, and if we add all 25 members, we get the correct answer 121392. But there is no need to perform this addition. By finding the twenty-fourth and twenty-fifth members, we can simply say that 46368, multiplied by 2, plus 28657 equals 121393, and then subtract 1.