CHAPTER III-FRACTIONS 43 nmon difference. The common difference multiplied by four the number of terms (in the required series) The sum a ated to these is the cube (of the chosen quantity) Examples in illustration thereof 28 The numerators begin with 2 and are successively increased 1; the denominators begin with 3 and are (also) successively creased by 1 and both these kinds of torms (umely, the merators and the denominators are (severally) five (in number). relation to these (chosen fractional quantities), give ont, 0 end, the cubic sum and the (corresponding) first terin, common fference, and number of terms The rule for finding out, from the known sum, first term and mmon difference (of a given series in arithmetical progression), e first term and the common difference (of a series), the optionally osen sum (whereof) is twice, three times, half, one-third, or some ch (multiple or fraction of the known sum of the given series) :- 29. Put down in two places (for facility of working) the osen sum as divided by the known sum This (quotient), when ultiplied by the (known) common difference, gives the (required) mmon difference and that (same quotient), when multiplied the (known) first term, gives the (required) first term-of (the ries of which) the sum is either a multiple or a fraction (of the 1own sum of the given series). Examples in illustration thereof 30. The first term (of a series) is, the common difference is and the number of terms common (to the given as well as the (2x)=3 The general applicability this process can be at once made out om the cquality, (pr) = r, so that in all such cases the number of terms the series is obtained by multiplying by the first term, which is representable X
- and the common difference is of course taken to be twice this first term
every case. 29. See note under 81, Chap. II.
