way of making 1, 2 of making 2; then there must be, by our rule, 2² ways of making 3, 2³ ways of making 4; and so on.
This table exhibits the ways of making 1, 2, 3, and 4. Hence it follows (which I leave the reader to investigate) that there are twice as many ways of forming a + b as there are of forming a and then annexing to it a formation of b; four times as many ways of forming a + b + c as there are of annexing to a formation of a formations of b and of c; and so on. Also, in summing numbers which make up a + b, there are ways in which a is a rest, and ways in which it is not, and as many of one as of the other.
Required the number of ways in which a number can be compounded of odd numbers, different orders counting as different ways. If a be the number of ways in which n can be so made, and b the number of ways in which n + 1 can be made, then a + b must be the number of ways in which n + 2 can be made; for every way of making 12 out of odd numbers is either a way of making
10 with the last number increased by 2, or a way of making 11 with a 1 annexed. Thus, 1 + 5 + 3 + 3 gives 12, formed from 1 + 5 + 3 + 1 giving 10. But 1 + 9 + 1 + 1 is formed from 1 + 9 + 1 giving 11. Consequently, the number of ways of forming 12 is the sum of the number of ways of forming 10 and of forming 11. Now, 1 can only be formed in 1 way, and 2 can only be formed in 1 way; hence 3 can only be formed in 1 + 1 or 2 ways, 4 in only 1 + 2 or 3 ways. If we take the series 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, &c. in which each number is the sum of the two preceding, then the nth number of this set is the number of ways (orders counting) in which n can be formed of odd numbers. Thus, 10 can be formed in 55 ways, 11 in 89 ways, &c.
Shew that the number of ways in which mk can be made of numbers divisible by m (orders counting) is 2ᵏ⁻¹.