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TO THE RESOLUTION OF PROBLEMS. 
113 
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PROBLEM LXII. 
Having given the sum fbj, and the sum o f the squares 
fcj of any given number of terms in arithmetical pro 
gression; to find the progression. 
Let the common difference be c, the first term T + e, 
and the number of terms n : then, by the question, we 
shall have 
* 4- e 4- x 4- 2e 4- x -f 3e x + ne — b, and 
x -f e| l -f- x 4- 2ep + a? 4- 3ej'...... a; 4- nej 1 =r c. 
But (by Sect. io, Theo. 4.) the sum of the first of these 
progressions is nx 4- ^ 1 ' C : And the sum of the 
second (as will be shewn further on) is r: nx 2 4- 
n . x 4- 1 . xe 4- therefore our 
two equations will become 
. ».«4-l.c , i 
nx 4- = b, and 
three 
to find 
br 
nx 1 4- n . n l . xe 4- 
n . n 4- 1 . 2« 4- 1 . e 
Let the former whereof be squared, and the latter 
multiplied by n, and we shall thence have 
nrx* + n . n 4- l . xe 4- 
ri 1 . n 4- 1 
. e‘ 
— b 1 , and 
iinber, 
rf the 
be № 
; and 
nation. 
nctei 
; ti 
) are 
■* a . . TT . n • n 4- 1 • 2n + 1 . e 
n x + n . n 4-1 . xe 4- - r= nc: 
let the first of these be subtracted from the second, so 
shall -iltLlg - ^±JLlL - b\ 
6 4 
But 
n % . n 4- l . 2 n 4- l n 2 . n f 1 f 
6 
is n n 2 . n 4' 1 X 
Qn 4-1 «4-1 
ri*. n 4- 1 x 
8;? 4- 4 — 64 — 6 __ 
24