| (99 + 1) | 99 | , or | 100 × 99 | , or 4950. |
| 2 | 2 |
The sum of 50 terms of the series
| 1 | , | 2 | , | 1, | 4 | , | 5 | , 2, &c. is | ( | 1 | + | 50 | ) | 50 | , |
| 3 | 3 | 3 | 3 | 3 | 3 | 2 |
or 17 × 25, or 425.
173. The first term being given, and also the common difference and number of terms, the last term may be found by adding to the first term the common difference multiplied by one less than the number of terms. For it is evident that the second term differs from the first by the common difference, the third term by twice, the fourth term by three times the common difference; and so on. Or, the passage from the first to the nth term is made by n-1 steps, at each of which the common difference is added.
EXERCISES.
| Given. | To find. | ||||
|---|---|---|---|---|---|
| Series. | No. of terms. | Last term. | Sum. | ||
| 4, | 6½, | 9, &c. | 33 | 84 | 1452 |
| 1, | 3, | 5, &c. | 28 | 55 | 784 |
| 2, | 20, | 38, &c. | 100,000 | 1799984 | 89999300000 |
174. The sum being given, the number of terms, and the first term, we can thence find the common difference. Suppose, for example, the first term of a series to be one, the number of terms 100, and the sum 10,000. Since 10,000 was made by multiplying the sum of the first and last terms by ¹⁰⁰/₂, if we divide by this, we shall recover the sum of the first and last terms. Now, ¹⁰,⁰⁰⁰/₁ divided by ¹⁰⁰/₂ is (122) 200, and the first term being 1, the last term is 199. We have then to pass from 1 to 199, or through 198, by 99 equal steps. Each step is, therefore, ¹⁹⁸/⁹⁹, or 2, which is the common difference; or the series is 1, 3, 5, &c., up to 199.
| Given. | To find. | |||
|---|---|---|---|---|
| Sum. | No. of terms. | First term. | Last term. | Common diff. |
| 1809025 | 1345 | 1 | 2689 | 2 |
| 44 | 10 | 3 | 29 | 14 |
| 5 | 45 | |||
| 7075600 | 1330 | 4 | 10636 | 8 |
175. We now return to (170), in which we compared two numbers together by their difference. This, however, is not the method of comparison which we employ in common life, as any single familiar instance will shew. For example, we say of A, who has 10 thousand pounds, that he is much richer than B, who has only 3 thousand; but we do not say that C, who has 107 thousand pounds, is much richer than D, who has 100 thousand, though the difference of fortune is the same in both cases, viz. 7 thousand pounds. In comparing numbers we take into our reckoning not only the differences, but the numbers themselves. Thus, if B and D both received 7 thousand pounds, B would receive 233 pounds and a third for every 100 pounds which he had before, while D for every 100 pounds would receive only 7 pounds. And though, in the view taken in (170), 3 is as near to 10 as 100 is to 107, yet, in the light in which we now regard them, 3 is not so near to 10 as 100 is to 107, for 3 differs from 10 by more than twice itself, while 100 does not differ from 107 by so much as one-fifth of itself. This is expressed in mathematical language by saying, that the ratio or proportion of 10 to 3 is greater than the ratio or proportion of 107 to 100. We proceed to define these terms more accurately.