—1 — 0; 
U-3B, 
•.or 
serai expression -— ——— (or -\—— 4- j 
found by case 2°, there will come out 385, for the re 
quired sum of the progression: which, the number of 
terms being here small, may be easily confirmed, by ac 
tually adding? theVo’terms together. Secondly, let it 
be required to imÿ/he number of cannon shot in a 
square pile whose side Is.5*0; then, by writing 50 for n 
in the same expression, 
n .11 \r I 
] 42925, expressing the number of 
shot in such a pile. Lastly, suppose a pyramid com- 
Ì )03ed of 100 stones of a cubical figure ; whereof the 
ength of the side of the highest is one inch ; of the 
second two inches; of the third three inches, &c. 
Here, by writing 100 instead of n, in the third general 
expression, we have 25502500, for the number of solid 
inches in such a pyramid. 
Hitherto regard has been had to such progressions as 
have unity for the first term, and likewise for the 
common difference; but the same equations, or theo 
rems, with very little trouble, may be also extended 
to those cases where the first term, and the common 
difference, are any given numbers, provided the for 
mer of them be any multiple of the latter. Thus, sup 
pose it were required to find the sum of the progression 
6* 4- 8 2 4- 10* &c. (or 36 4 : 64 -f 100 &c.) conti 
nued to eight terms : then, by making (4), the square of 
the common difference, a general multiplicator, the given. 
expression will be reduced to 4 x 3‘+4 i + 5 I ....io*: 
but the sum of the progression 1* 4- 2 2 4- 3 2 4- 4 2 ... io l 
is found, by the second Theorem, to be 385 ; from 
which, if (5), the sum of the two first terms, (which the 
series 3 2 4- 4 2 4- 5 2 10 2 wants,) be taken away, 
the remainder will be 380 ; and this, multiplied by 4, 
gives 1520, for the true sum of the proposed progres 
sion : and so of others. 
But if the first term is not divisible by the common 
difference, as in the progression, 5 l 4- 7* 4- q 2 Sec. 
the speculation is à little more difficult; nevertheless».