________________
CHAPTER VI-MIXED PROBLEMS.
171
product is divided by sir. (To this resulting quotiont), the square of the first term and tho (continued product) of the number of terms as diminished by one, the first term, and the common differenoe, are added. The whole of this multiplied by the number of torms becomes the required result.
. Examples in illustration thereuf. 300. (In a series in arithmetical progression), the first term is 3, the common difference is 5, the number of terms is 5. Give out the sum of the squaros (of tho torms) in the series. (Similarly, in another series), 5 is the first torm, 3 the common difference, and 7 the number of terms. What is tho sun of the squaros (of the torms) in this series?
The rule for arriving at the sum of the uubos (of a givo number of natural numbers) :: -
301. The quantity represented by the square of half tho (given) number of torms is multiplied by the wuare of the sun of one and the number of terms. Tu this (nience of) arithmetic, this result is said to be the sum of the cubos (of the given number of natural numbers) by thoso who know the socrot of oalculation.
Erumples in illustration thereof. 302. Give out (in ench case the sum of the cubes of (tho natural numbers up to) 6, 8, 7, 25 and 250.
The rule for arriving at the sum of the cubes (of the torme in a series in arithmetical progression), the first torm, the commou difference, and the number of terms wkeroof aro optionally chosen :
303. The sum of the simplo terms in the given sorios), as multiplied by the first torm (therein), is (further) multiplied by the
301. Algebraically
=), which is the sum of the cubes of the natural numbers up to n.
303. Algebruioally, tva (a b) + b = the sum of the cubox of the terms in a series in arithmetical progression, where I = the sum of the simple tornis of the series. The sign of the first tertu in the formula is + or - according as .> or <b.
