| 1 | 2 | 3 | 4 | 5 |
| 2 | 4 | 8 | 16 | 32 |
If the third and fourth terms, 8 and 16, be multiplied together, the product, 128, will be the seventh term of the series. In like manner, if the fifth term be multiplied into itself, the product will be the tenth term; and if that sum be multiplied into itself, the product will be the twentieth term. Therefore, to find the last, or twentieth term of a geometric series, it is not necessary to continue the series beyond a few of the first terms.
Previous to the numerical recreations, we shall here describe certain mechanical methods of performing arithmetical calculations, such as are not only in themselves entertaining, but will be found more or less useful to the young reader.
To Find a Number Thought of.
FIRST METHOD.
| EXAMPLE. | |
| Let a person think of a number, say | 6 |
| 1. Let him multiply by 3 | 18 |
| 2. Add 1 | 19 |
| 3. Multiply by 3 | 57 |
| 4. Add to this the number thought of | 63 |
Let him inform you what is the number produced; it will always end with 3. Strike off the 3, and inform him that he thought of 6.
SECOND METHOD.
| EXAMPLE. | |
| Suppose the number thought of to be | 6 |
| 1. Let him double it | 12 |
| 2. Add 4 | 16 |
| 3. Multiply by 5 | 80 |
| 4. Add 12 | 92 |
| 5. Multiply by 10 | 920 |
Let him inform you what is the number produced. You must then, in every case, subtract 320; the remainder is, in this example, 600; strike off the 2 ciphers, and announce 6 as the number thought of.