and met the waggon just40 minutes before he came to the
thirty first milestone. How far used to be B from L when A reached it?
Fifty eight EQUATION PAPERS.
1. I(5x-l)- A"(7x-2) = 6J-stock of corn, expectingto sell it in
6 months at 3s a bushel more tha
335. There are a n* of tickets,marked with some some of the bneumr-s
from 1 to n'+ 1. Each this sort of numbers, (such
as r,)is marked upon the same quantity (r) of tickets ; and
every ticket marked with a square number m^) confers a
prizeof m shillingsA.
x (g,fifty four MISCELLANEOUS EXAMPLES.
344. If tty h,Cf "c. Are any n portions, shew that "+ 5*+c"+ "c
"n aho .);and thence show that 1.2.3 .,n "^ (n+ l)
345. A playsat a game, wherein he reckons his risk of success
to be e. If it be a fair danger that h
1* 2" 3*
!" 2" three"
!" 2" 3"
^ 2+2-^+23^-' 3
+ 3-i+33 + "c., ^-^, + _-"c.
328. What number of triangular pyramidscan be fashioned, whose edges
are six givenlines,any two of which can be " than the 1/3 ?
329. Do away with m, w, jt;,q from the equatio
819. Investigattehe generalforms of x and y, which rationalize
ax^ 4- hy*^ywhen w = 3 or any bizarre quantity.
320. Fifteen folks sit down at a round desk. Shew that it's
6 to 1 againsttwo particulaprersons sittinngext every different;
a ^a-y)x^(a - 2y) a:*four- "c. Ad inf, *
and if x obtain values in h.P., shew that the corresponding
values of y will likely be in A. P.
259. Shew that the series x - " " " + - "c. Converges
at the w^h time period, if n"x and find.
260. There are th
308. If z" + a + 1 = zero, shew that the sum of those phrases of the
expansionof (1 f a;)", in which the index of a; is a mult, of 3,
= J (1+ xzf + (1 + xy four- (1 + xz'r.
2 three four
309. Sum to n phrases and ad infinitumr-^-r + ^-77:
2d half a consistent ratio. Shew that there is however one
245. Compare the chances of throwingtwo aces onlyin two trials
with three dice and in three trialswith 4.
246. If 2a;,= y^'*^"=' - y^'-^'t^h'en, x, (x^+ a;,+ "c. + x^,^,=)x"*,
coin is taken from one (it's not identified which) and
droppedinto the other; and then, on drawing a coin from
every bag, theyare observed to be two shillings.What's the
chance that this may occasionally arise once more,if two extra are drawn,
one from ea
ax*+ with the aid of*+ cz'=)(a^3 + 6^+ c^)ok^.
232. Shew that,if pq = r, the equationa^ + px^ + ja; + r = 0 will
have two roots equaland of oppositesigns.
233. In finding three numbers with primeden", whose sum shallbe 1 ifs*
234. Resolve into partial fra
219. Sum to n terms
J tossinga coin,is to pay B is if it fall heads the firsttime,
25 if the 2d,3s if the 0.33,etc for n throws, the
sport to cease as quickly as itfallsheads. In finding ^*s expectation,
221. Get rid of a;,y, z, from the equations
207. Do away with a and h from the equations
208. S^yS^,S^y"c. Are the sums, to n phrases, of n Geom. Sequence^
whose firstterms are every team spirit,and fashioned ratios,1,2,three,
"c.: shew that
^j+ ^,+ 2^3+ three/S;+ "c. + (w-l)^"=r-i-2"+ three" + "c.
192. K flj: cfg: : a, : "three : : flg: cf^"c., then
("!*+ fl,+*"C.)(rtg+*^three*+ "C.)= ("!",+ rtgfl+fj"c.)'.
193. Shew that (1+ xj'+ (I - xf)v (1+ a:)+ V(l + a:)= 1 - a long way,
nearly,when a: is small: and to find its approximate value
when X is Zar^e
176. LiA^B, J^x^C, and Coc^Z^+^ZB*, shew that
177. Shew that abc"2a"h)1h"c)2c"a), except "=i=c.
178. A bequeathsto h"fk1st baby an n^ of his property +"P, to
the 2nd an v-^of th"the rest +"2P, and many others. They all
share alike : how many have been the
132. Symbolize^/2n^-l) in the form of a binomial surdj and
shew also that V(four+ three V- 20) + V(4 - 3 ^/- 20) = 6.
133. Given log If = .1461280, log.A hundred and forty four = 1.1583625,
and log.0441 = 2.6444386, in finding the logsof the 9 digits.
145. Divide unityinto 4 partsin A. P., so that the sum of their
cubes may be ^q,
146. Shew that (aJ),+ a,+ "c.)'" (a*-fa* + "c.)b*+ b* + "c.),
unless flj: tj= a^ : b^- "c.
147. Find the coefiicientof x* in (a + 5ar+ ex*)c"*.
148. Given A, my sales, a the
and what should J? Have staked, that -four's drawing first
could givehim no competencies?
121. In finding the n" of divisors of a hundred and forty, and the n" of numbers less
than a hundred and forty and high to it: express generallythe rational
a^-x- V("^ - ^ ) ^ c^ a^- a*
110. The veracities of A, B, C, being f, J, f, C asserts that
i've gained a prizeof "10 in a specific rafile,where there
are 10 tickets,of which I preserve two: what's the value of my
expectationi,f I in finding that C has onl
squares of the terms = productof sum of all the phrases through
excess of the atypical terms above the even.
Ninety six. If the G. Imply between x and y : the H. Imply :im :n, then
X : y i: m + v'(m*- w*): m - V(m*- w*).
97. Find the sum of the entire num
. 80. A had in his pocket a sovereignand 4 shilling;staking
out two cash at random, he promisesto provide them to B and C:
what's the valued at of C"s expectation?
Eighty one. Shew that (" + " - cf + (6 -f c - ctf+ (c + a - 6)'" Zabc.
Eighty two. El
be the Arithmetic mean between m and w, and the Geometric
mean between a and b.
MISCELLANEOUS EXAMPLES. 35
66. Find the coefficients of x'^ and x* in (a - 5x* "]"ex - dx~^Y"
sixty seven. The mail from A begins for B at p hrs P.M., and that from B
for A oi
(^a+b+c+df = (a+")H(c+e^2^2(5+c)2.
381. The No. Of Var"^ of n thingsr,together: the No., r-1 gteo-ther
: : 10 : 1,and the correspondinNgos. Of Comb"^ are as
5:three; find n and r.
382. A character makes 20 lbs. Of tea at 4s. 9J.,with the aid of mixing thr
taken from each to make 5 gals,of wine and 9 of water ?
18. Obtain the square root of 1 + (1- c*)"* in the form a* + /3^19. Solve the equation H = -.
A + a: *J(a^ x) x
20. A flag-sta(fc)fstands at the top of a tower, whose heightis a.
Find the distance fr
= 2 (a- 6) (rt- c) + (6 - a) (6 - c) + (c - a) (c- 6).
4. A starts from a certain place,and travels a miles the first day,
2a the second, 3a the third, "c.; after 4 days, B starts to
overtake him, travelling9a miles per day. After how many
days will he co
and if 2^. Be taken from ^'s reward sum and added to j5's,
the latter amount will probably be of the previous. What had they
every at first?
367. To find the worth of s/a-6axndysquare the outcomes.
368. If the differenceof two fractions be mn- show that 7
end sixteensooner than he otherwise would : what time
will he take,if he onlybeginto quickenhis % halfway?
E349. Divide (^-l)a"-(:r3+^2_2)a24-(4A^H3.'2r)-af-3(^+l)
350. MultiplyA^a'^ VKc)Hya/"^-v/(A)2.
351. If x=^i/'-ir-'Vr'-ijq^
A " n.A"ln a " Sn,^
320. If the sum or change of two numbers be 1,exhibit that the
change of their squares is the change or sum of the
321. A servant agreedto reside along with his master for "S a 12 months and
numbers is divisibleby three times the middle number.
284. If a : 6 : : c : e7,show that ^a" - W : 2^ ~ 3^?^: : aH^' : cH^285. Two-thirds of a designated quantity of bad folks obtained
Is. Qd, each and every,and the rest 2s. 6c?. Each and every : the whol