601. In arithmetical progression there are five parts to be
considered, viz : the first term, the last term, the number of terms,
ike common difference, and tlie sum of all the terms. These parts
have such a relation to each other, that if any three of them are
given, the other two may be easily found.
G02. If the sum of the two extremes of an arithmetical pro-
gression is multiplied by the. number of the terms, the product
will be double the sum of all the terms in the series.
Take the series 2, 4, 6, 8, 10, 12.
The same inverted 12, 10, 8, 6, 4, .2.
The sums of the terms are 14, 14, 14, 14, 14, 14.
Thus, the sum of all the terms in the doutle series, is equal to
the sum of the extremes repeated as many times as there are terms;
that is, the sum of the double series is equal to 12+2 multiplied
by 6. But this is twice the sum of the singk series. Hence,

603. To find the sum of all the terms, when the extremet
and the number of terms are given.
Multiply half the sum of the extremes by tlie number of terms,
and the product will be the sum, of the given series.
Obs. The reason of this process is manifest from the preceding illustrition.
Ex. 1. The extremes of a series are 3 and. 25, and the number
of terms is 12 : what is the sum of all the terms ? Ans. 168.
