Elements of Algebra

CHAP. V. OF ALGEBRA. 139 gression, having $10$ for its first term, $\frac{1}{4}$ for the common difference, and the number of terms $21$. Ans. $140$. 3. Required the number of all the strokes of a clock in twelve hours, that is, a complete revolution of the index. Ans. $78$. 4. The clocks of Italy go on to $24$ hours; how many strokes do they strike in a complete revolution of the index? Ans. $300$. 5. One hundred stones being placed on the ground, in a straight line, at the distance of a yard from each other, how far will a person travel who shall bring them one by one to a basket, which is placed one yard from the first stone? Ans. $5$ miles and $1300$ yards. *** CHAP. V. *Of Figurate*, or Polygonal Numbers. 425. The summation of arithmetical progressions, which begin by $1$, and the difference of which is $1, 2, 3$, or any * The French translator has justly observed, in his note at the conclusion of this chapter, that algebraists make a distinction between figurate and polygonal numbers; but as he has not entered far upon this subject, the following illustration may not be unacceptable. It will be immediately perceived in the following Table, that each series is derived immediately from the foregoing one, being the sum of all its terms from the beginning to that place; and hence also the law of continuation, and the general term of each series, will be readily discovered. Natural $1, 2, 3, 4, 5$ - $n$ general term Triangular $1, 3, 6, 10, 15$ - $\frac{n.(n+1)}{2}$ Pyramidal $1, 4, 10, 20, 35$ - $\frac{n.(n+1).(n+2)}{2.3}$ Triangular-pyramidal $\} 1, 5, 15, 35, 70$ - $\frac{n.(n+1).(n+2).(n+3)}{2.3.4}$ And, in general, the figurate number of any order $m$ will be expressed by the formula, $\frac{n.(n+1).(n+2).(n+3) \ldots (n+m-1)}{1.2 \cdot 3 \cdot 4 \ldots m}$ Now, one of the principal properties of these numbers, and