What is the infinite geometric series of which the sum is 2 and the second term

?

7. If a straight line be divided into two parts so that the rectangle contained by the whole line and one part is equal to six times the square of the other part; find the ratio of the parts.

EXERCISE LXII.

1. The product of two expressions is

(x+y)3+3xy (1-x-y)-1,

and one is x+y-1, find without division the other, explaining the method you adopt.

4. If a and B are the roots of the equation

x2+px+9=0,

shew that a3 + ß3=3pq-p3, and find the value of

1 1
+

a3 B3

5. If a b c d, shew that

(i) a+mb :c+md :: a−mb : c-md,

(ii) a a+b: ac-bc: ac-bd.

6. Write down the first four terms of the series whose nth term is 2+ (−1)′′n.

Find what number of terms of the series 6+9+12+... will amount to 105, and of the series 13+ 10 +7 + ... will amount to 34.

7. In order to resist cavalry a battalion is usually formed into a hollow square, the men being four deep, but a single company is usually formed into a solid square. If the hollow of the square of a battalion, consisting of seven equal companies, is nine times as large as one of its companies' squares, find how many men there are in a company, assuming every man to occupy the same space.

EXERCISE LXIII.

1. Prove that x-na"-1x + (n-1) a" is divisible by (x-a)2 if n be a whole number, and x7-a7 by x2 +pax+a3 if p3 — p2 −2p+1=0.

and

2. Find the G.C.M. of

(ax+by)-(a-b) (x + z) (ax+by)+(a - b)2 xz,

(ax-by)-(a+b) (x + z) (ax-by)+(a+b)2 xz.

(iii) 2x+3y=24 in positive integers.

5. If the square of a vary as the cube of y, and x=2 when y = 3, find the equation between x and y.

6. If a, b, c, d be in Harmonical Progression, prove that the Harmonic mean between a and b is to that between c and d :: 3b-c: 3c-b.

7. The two sides of a rectangle, expressed in feet, have the sum of their cubes equal to 109 times their sum, and the difference of their cubes equal to 229 times their difference: find the area of the rectangle and its diagonal.

1. Shew that

EXERCISE LXIV.

(a+b)2+(a+c)2+(a+d)2 + (b+c)2 + (b + d)2 + (c+d)2

4. If x have to y the duplicate ratio of x+z to y+z, prove that z is a mean proportional between x and y. 5. If a, b, c be in continued proportion, prove that

(ii) (a2+b2) (b2+c2)=(ab+bc).

6. Eliminate x and y from the equations

x+y=a, x2+y2 = b2, x3+y3=c3.

7. Three men A, B, C are candidates for an office. If all had demanded a poll, the number of votes for them would have been in Arithmetical Progression, and A would have been elected by a majority equal to the number of voters for C, who has the fewest votes. C however withdraws before the election, and his supporters distribute their votes between A and B in the ratio of 1 to 4; thus A is elected by a majority of 40. Find the number of the electors.

EXERCISE LXV.

1. Divide

x-(a+b+c+d) x3 +(ab+ac+ad+be+bd+cd) x2

― (bcd+cda+dab + abc) x + abcd

by x2-(a+b)x+ab;

also "-1 by "-1.

2. Express in their simplest forms

2x2-x+2 4.002-1

4x3+3x+2 2x-1

(i)

X

3. Find a number such that when it is divided into any two parts a and b, a2+b shall always be equal to a+b2.