A treatise os Algebra.' 18/

be substituted for n, in the general Expression

” * ” 1 * ln - {or- 1 - 1 -— j found by Case z®,

and there will come out 385 for the required Sum ofthe Progression: Which, as the Number of Terms ishere small, may be easily confirmed, by actually addingthe 10 Terms together. Again, suppose it were requir-ed to find the Number of Cannon-shot, in a square Pilewhose Side is 50; then, by writing 50 instead of n in

n. n + 1 . zw-J- 1

the second general Expression---- we shall

50x51x1011(- 6 - )

42925, expressing the Number of

Shot in such a Pile. Lastly, suppose there were a Pyra-mid composed of 100 Stones of a cubical figure, whereofthe Length of the Side of the least is one Inch; of thesecond two Inches; of the third three Inches, b'c.Then, by writing 100 instead of n, in the third generalExpression, we shall have 25502500 for the Number offolid Inches in the Pyramid.

Hitherto we have had regard to such Progressions ashave Unity for their first Term, and likewise for thecommon Difference of their Roots; but the fame The-orems, with very little Trouble, may be also extendedto those Cases where the Root of the first Term, andthe common Difference are any given Numbers, providedthe former of them be any Multiple of the latter. Thus,suppose it were required to find the Sum of the Progres-sion 6 2 -f 8 2 + 10*, bfc. (or z6 64 -s- 100, &c.) con-tinu’d to 8 Terms: Then, by making (4.) the Squareof the common Difference a gener al Multiplier, the give nExpression will be reduced to 4. x 3 2 -f-4 a +5 2 .... 10 2 :

But the Sum of the Progression r 1 +z I +3 1 -f-4 2 .ro*

is found, by the second Theorem, to be 385; fromwhich if (;) the Sum of the two first Terms (which the

Series 3 2 -f- 4 2 -f- 5 2 .io 2 wants) be taken away,

the Remainder will be 380, and this, multiply’d by 4,gives 1520 for the true Sum of the Progression ; thelike of others. But