Elements of Algebra

136 ELEMENTS SECT. III. equal terms is $10$, as in the progression, and their sum consequently is equal to $10 \times 31 = 310$. Hence, as this sum is twice the sum of the arithmetical progression, the sum required must be $155$. 416. If we proceed in the same manner with respect to any arithmetical progression, the first term of which is $a$, the last $z$, and the number of terms $n$, writing under the given progression the same progression inverted, and adding term to term, we shall have a series of $n$ terms, each of which will be expressed by $a + z$; therefore the sum of this series will be $n(a + z)$, which is twice the sum of the proposed arithmetical progression; the latter, therefore, will be represented by $\frac{n(a+z)}{2}$. 417. This result furnishes an easy method of finding the sum of any arithmetical progression; and may be reduced to the following rule: Multiply the sum of the first and the last term by the number of terms, and half the product will be the sum of the whole progression. Or, which amounts to the same, multiply the sum of the first and the last term by half the number of terms. Or, multiply half the sum of the first and the last term by the whole number of terms. 418. It will be necessary to illustrate this rule by some examples. First, let it be required to find the sum of the progression of the natural numbers, $1, 2, 3$, &c. to $100$. This will be, by the first rule, $\frac{100 \times 101}{2} = \frac{10100}{2} = 5050$. If it were required to tell how many strokes a clock strikes in twelve hours; we must add together the numbers $1, 2, 3$, as far as $12$; now this sum is found immediately to be $\frac{12 \times 13}{2} = 6 \times 13 = 78$. If we wished to know the sum of the same progression continued to $1000$, we should find it to be $500500$; and the sum of this progression, continued to $10000$, would be $50005000$. 419. Suppose a person buys a horse, on condition that for the first nail he shall pay $5$ pence, for the second $8$ pence, for the third $11$ pence, and so on, always increasing $3$ pence more for each nail, the whole number of which is $32$; required the purchase of the horse? In this question it is required to find the sum of an arithmetical progression, the first term of which is $5$, the difference $3$, and the number of terms $32$; we must there-