Online Encyclopedia


Online Encyclopedia
Originally appearing in Volume V10, Page 335 of the 1911 Encyclopedia Britannica.
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FIGURATE NUMBERS, in mathematics. If we take the sum of nterms of the series 1+1+1+ ..., i.e. n, as the nth term of a new series, we obtain the series 1+2+3+ . . ., the sum of n terms of which is a n . n+r. Taking this sum as the nth term, we obtain the series 1+3+6+1o+ ..., which has for the sum of n terms n (n+1) (n+2)/3! 1 This sum is taken as the nth term of the next series, and proceeding in this way we obtain series having the following nth terms: 1, n, n(n+1)/2!, n(n+1) (n+2) /3!,...n(n+1) ... (n+r—2)l(r— 1) !. The numbers obtained by giving n any value in these expressions are of the first, second, third, . . . or rth order- of figurate numbers. Pascal treated these numbers in his Traite du triangle arith- metique (1665), using them to develop a theory of combinations and to solve problems in proba- t t t t - . , j r bility. His table is here shown pO©O in its simplest form. It is to be noticed that each number is the sum of the numbers immediately above and to the left of it; and that the numbers along a line, termed a base, which cuts off an equal number of units along the top row and column are the coefficients in the binomial ex- pansion of (t+x)'-1, where r represents the number of units cut off.

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