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Ratios, AND PROPORTION. (16.) Ratios. Principles.
If a >b, then
a + x : 6+ x > or <a : b, according as a < or > b.
and generally, ma + nb : pa + qb :: mc+nd : pc+qd.
(W. 171-182; E. 461-5.) If a : 6 :: C:d :: e: f
:: &c. then
If a : 6 :: C : d,
and e:f:: g: h; then
(W. 184–8.) If two numbers be prime to each other they are the least in that proportion.
(W. 189.) Compound proportion.
(18.) Let am represent the moth term of an arithmetical series,
b, the common difference,
To insert n arithmetical means between c, and e.
Given the pth and oth terms, to find the oth term.
(19.) Any three of the quantities az, an, b, n, s; being given, the others may be found from the following formulæ :
HARMONICAL PROGRESSION. (20.) The reciprocals of quantities in arithmetical progression are in harmonical progression, and conversely. To insert n harmonical means between c, and e. The mth mean =
(n-m + 1)e+mc Given the pth and qth terms, to find the poth term. (q-pap.aq
(W. 381.) (r-pa,-(-9)
FIGURATE, AND POLYGONAL NUMBERS.
(21.) The nth term of the mth order of figurate numbers = the sum of n terms of the (m- 1)th order
n 7-min + m 1......N+1.
m .m +1
« The above proposition may be thus compendiously stated:
For an explanation of this notation see the Appendix.
The nth term of the series of m-gonal numbers is the sum of n terms of the arithmetical progression
1, 1+ m - 2, 1 + 2.m-2, &c.
— m – 4. n}. The sum of the reciprocals of n figurate numbers of the mth order
1 m-1 1.2.3...(m-1) (n+1) (n + 2)...(n + m - 1)
(G. A. 132, 3; L. C. 91, 2; E. 425–39; B. 27, 8.)
On-m: @ne + m=(an) .
The mth mean=c
(W. 214–22; E. 505-22; Bour. 199 208.)