713. If a, b and c be in geometrical progression, determine which is greater, a2+b2+c2 or (a−b+c)2.

715. If a, b, c, d are in geometrical progression, prove that (a+b+c+d)2= (a+b)2 + (c + d)2+2(b+c)2.

716. In a geometrical progression, if P and Q denote the pth and qth terms, find the nth.

717. If s = sum of the squares of any two quantities, p = twice their product, and P = the pth power of the sum, prove that P.P.p.p... to infinity =s"+p2.s"-1+p3‚P—1.sp−2

-I

I

718. Given s, and s2, the sums of the even and odd terms of a geometrical progression of 2n terms. Of m arithmetic means inserted between its pth and qth terms, required the rth mean.

719. If P be the product, S the sum, and S, the sum of the reciprocals of n quantities in geometrical progression, prove

720. The sum of £700 was divided among four persons, whose shares were in geometrical progression; and the difference between the greatest and least was to the difference between the means as 37 to 12. What were their respective shares?

HARMONICAL PROGRESSION.

General Formulæ. If a, b, c, d, . . be consecutive terms of a series in harmonical progression, then will

a:c=a-b: b- c and b: db-c: c―d, &c.;

I I I I

also

&c. are in arithmetical progression.

721. Continue in both directions the series 2, 3, 6, and 3, 4, 6. How far can the latter be continued either way?

722. Insert two harmonic means between 2 and 4. 723. Insert two harmonic means between 6 and 24. 724. Insert four harmonic means between 2 and 12. 725. Insert six harmonic means between I and 20. 726. Insert n harmonic means between x and y.

727. Given a and b, the first and nth terms respectively of an harmonic series; find the intervening terms.

728. Given x, y, the first and second terms of an harmonical progression; continue the series, and write down the nth term. 729. Show that the arithmetic, geometric and harmonic means between a and b are in continued proportion.

730. If a, b, c are in harmonical progression, determine which is greater, 262 or a2+c2.

731. Having given the mth and nth terms of an harmonical progression, viz. M and N respectively, determine the (m+n)th term. 732. There are four numbers of which the first three are in arithmetical progression, the last three in harmonical; prove that the first is to the second, as the third is to the fourth.

733. If y be the harmonic mean between x and z, and ≈ and z be the arithmetic and geometric means respectively between a and b; express y in terms of a and b.

734. If a, b, c be the pth, qth and rth terms respectively of an harmonic series, prove that (p-q)ab+(r−p)ac+(q-r)bc=0. 735. If a*=by=c=&c. and a, b, c, &c. be in geometrical progression, then will x, y, z, &c. be in harmonical progression.

736. Compare the lengths of the sides of a right-angled triangle when the squares described upon them are in harmonical progres

sion.

737. Having given II, the sum of three numbers in harmonical progression, and 36 their continued product; determine the numbers.

THE PILING OF BALLS AND SHELLS.

General Formulæ. The number of balls in a triangular pile is expressed by 'n(n+1)(n+2), and in a square pile by

I

n(n+1)(2n+1); where in both cases n denotes the number of balls in one side of the bottom row. Also the number of balls in a rectangular pile is represented by n(n+1)(3l−n+1), 6

I

where l is the number of balls in the length of the base row, and n the number in the breadth of the same row.

738. Show that 13 + 23 + 33 + . . . + n3 = (1 + 2 + 3 + ... + n)2. 739. Find the number of balls in a triangular pile, each side of the base row containing 36 balls.

740. Find the number of shells in a square pile, each side of the base row containing 32 shells.

741. Required the number of balls in a rectangular pile, the length and breadth of the base row being 52 and 34.

742. What is the number of balls in an incomplete triangular pile, in which 25 balls form one side of the bottom row, and 13 a side of the top row?

743. What is the number of balls in an incomplete square pile, one side of the bottom containing 44 balls, and of the top 22?

744. An incomplete rectangular pile of 18 courses contains 56 balls in the length, and 38 in the breadth of the bottom row: required the number of balls piled.

745. There are two complete piles of balls, one triangular, the other square; such that the whole number of balls in the former is to the whole number in the latter as 6 is to II. Required the number of balls in each pile, the bottom row of both having the same number of balls in a side.

746. A complete rectangular pile has the same number of balls in the breadth of its bottom row as form the side of the base of a complete square pile; and the number of balls in the former pile is to the number in the latter as 8 is to 5. Also the number of courses in the rectangular pile added to the number of balls in the length of its bottom row is 29. Required the whole number of balls in each pile.

747. What is the number of shells contained in 7 courses of an incomplete pentagonal pile, when each side of the base row contains 15 shells?

PERMUTATIONS AND COMBINATIONS.

General Formulæ.

1. The number of permutations of n different things, taking r of them at a time, is n(n-1)(n-2)...(n−r+1).

2. The number of permutations of n things, where p of them are of one sort, q of another, r of a third, and so on, all taken

together, is

1.2.3...n

(1.2.3..p)(1.2.3..q)(1.2.3..r)(&c.)

3. The number of combinations of n different things, taking r of them together, is n(n-1)(n-2)...(n−r+1). I.2.3.....r

748. If ten letters, a, b, c... be combined, 5 and 5 together, in how many of the combinations will a and b occur?

749. How many different sums can be formed with the following coins: a farthing, a penny, a sixpence, a shilling, a crown, a halfsovereign, a guinea, and a moidore?

750. At an election where every voter may vote for any number of candidates not greater than the number to be elected; there are 4 candidates and 3 members to be chosen, in how many ways may a man vote?

751. From a company of 80 men, 9 are draughted off every

night to form a patrol; on how many different nights can a different selection be made; and on how many of these will any particular soldier be engaged?

752. Find all the permutations that can be formed out of the letters of the words (1) Baccalaureus, (2) Mississippi, (3) Hippopotamus, (4) Museum, (5) Commencement, (6) Couscoussou.

753. In how many different ways may the letters of the continued product a5.c.ef be written?

754. How many combinations will there be of 12 letters, a, b, c, &c., taken 4 and 4 together? In how many of these will a, b and

c occur?

755. At a game of cards, 3 being dealt to each person, any one can have 425 times as many hands as there are cards in the pack; required the number of cards.

756. The number of combinations of n things taken 4 at a time is to the number where 2 are taken at a time as 15 to 2. Required the value of n.

757. The number of permutations of n things taken 3 at a time is equal to 6 times the number of combinations of n taken 4 at a time; find n.

758. The number of combinations of n things taken 5 at a time is to the number where 3 are taken at a time as 18 is to 5; find n. 759. The number of permutations of n things taken r at a time is equal to 10 times the number taken (r−1) at a time: and the number of combinations of n things taken r at a time is to the number taken (-1) at a time as 5 is to 3; required the values of n and r.

760. If P2, P3... Pn represent the number of permutations that can be formed out of n quantities taken 2, 3, &c. n together respectively, and P=P2P3... Pn, prove that

P=P3Pn{(P3—P2)(P4—P3)(P5—P4) · · · (Pn−x—Pn−2) } . 761. How many words can be formed consisting of 3 consonants and a vowel, in a language which contains 19 consonants and 5 vowels?

762. Find the number of words which can be formed by taking the 24 letters of the alphabet, 6 at a time, each word containing 2 vowels.

763. Out of 17 consonants and 5 vowels, how many words can be formed, each consisting of 2 consonants and I vowel?

764. The total number of combinations of 2n things divided by the total number of combinations of n things is equal to 65; find n. 765. There are 4 regular polyhedrons marked in the manner of dice, and the numbers of their faces are 3, 6, 8, 12 respectively; taking all of them together, how many different throws can possibly be made?

766. The number of combinations of n things taken p together is to the number taken p+2 together as a is to b; if a, b, p represent 5, 18, 3 respectively, find n.

767. Find the number of permutations of n things taken r at a time, with repetitions, i. e. allowing quantities which recur to be combined as if they were different.

768. Find the number of different combinations of n things taken 1, 2, 3 ... n together; of which n things, p are of one sort, q of another, r of a third, and so on, subject to the condition that p+q+r+. .=n.

769. When the number of combinations of 2n things taken r at a time is the greatest possible; find r.

770. Express the number of combinations of n+1 things taken r at a time, in terms of the number of combinations of n things taken r at a time, and also r- I at a time.

771. Investigate a formula for expressing the number of homogeneous products which can be formed of n things of r dimen. sions; and determine the number of terms in the expansion of (a+b+c+d)1o.

is positive or negative, and integral or fractional.

In the expansion of (a+be+cx2+...+kat)n the general term is represented by

1.2.3. ..n

(1.2.3...p)(1.2.3..q)(1.2.3...r) &c.

-ar.ba.cr,&c. x1+2r+.....,

where p+q+r+..=n, and q+2r+...=m, if m denote the index of a in the general term.

772. Expand (a+x), (a−x)7, (2x-3y)4, and (5-8)°.

4

773. Expand (1+2)", (1–2)2, (a+12)3, and (~-—3y)3.

3

x

774. Expand (c-x)−', (a+h)-, (1−x)—a, (a2—x2)−3, and (a+x)-2.

775. Expand (a+∞ √—1)”, (b—y ✔ — 1) ̊, and (h+k √ −1)',