5. 12 cwt., 30 lbs.; 3 cwt., 30 lbs.; 2cwt., 50lbs. 6. £396; £324. 7. 1116-7744 lbs. of oxygen, 969 136 lbs. of carbon, 154-0896 lbs. of hydrogen. 8. £3250; £2166. 13s. 4d.; £1083. 6s. 8d. 10. £12. 10s., £12. 10s., £25, £50. 9. £350; £450. 11. A's share = £5000, B's share= £3750, C's share= £3125. APPENDIX TO ALGEBRA. ART. 154. ARITHMETICAL PROGRESSION. DEFINITION. IN Algebra a set of quantities, which mutually depend upon each other according to some fixed law, is called a SERIES. DEF. If a series of quantities taken in order, increase or decrease by the addition or subtraction of the same quantity, which is called the COMMON DIFFERENCE, such quantities are said to be in ARITHMETICAL PROGRESSION. Thus, the series of numbers 1, 2, 3, 4, 5, 6, &c. which increase by the addition of 1 to each successive term, or in which series each term is greater than the one which precedes it by the number or common difference 1, forms an Arithmetical Progression. Again, the series of numbers 19, 17, 15, 13, 11, &c. which decrease by the subtraction of 2 from each successive term, or in which series each term is less than the one which precedes it by the number or comnon difference 2, forms an Arithmetical Progression. Similarly, each of the following series 6, 3, 0, +3, +6, +9, &c.; 0, 4, -2, -8, -14, &c.; and 2x, 5x, 8x, 11x, &c. forms an Arithmetical Progression, the common difference in the first series being + 3, in the econd - 6, and in the third + 3x. The general form of a series in Arithmetical Progression is a, a + b, a+2b, a+3b, a+ 4b, &c. or a, a-b, a-2b, a-3b, a-4b, &c. the former the quantities go on increasing, and in the latter decreasing the common difference b. 155. In the series a, a+b, a+2b, a+3b, &c. the first term is a, which =a+0× b=a+(1−1) b, the second term is a+b, which =a+1×b=a+(2−1) b, the third term is a + 2b, which=a+2xb=a+(3−1) b, hence it appears, that the coefficient of b in any term is always less by unity than the number which denotes the place of that term in the series; .. the (n-2)th term will=a+{(n−2)−1}b=a+(n−3) b, the (n-1)th term will=a+{(n-1)-1}b=a+ (n-2) b, and the nth term will a+(n−1) b. Also, in the series a, a-b, a-2b, a-3b, &c. the first term is a, which = a+0× −b=a¬0×b=a-(1-1)b, the second term is a-b, which =a+(2−1) × −b=a−(2—1) b, the third term is a−2b, which =a+(3−1) × −b=a− (3—1) b, and so on; -b the nth term will be a+ (n-1) × – =a-(n-1) b, which is the same as the nth term of the first series, with the exception of -b being substituted for+b. 156. To find any term of a series in Arithmetical Progression whose first term, and common difference are known. Ex. 1. Find the 25th term of the series 2, 5, 8, 11, &c. In the above formula a=2, b=3, n= Ex. 2. Find the 10th term of the series 15, 11, 7, 3, &c. In the above formula a=15, b=-4, n=10; .. 10th term = 15+ (10−1) × −4 =15-9x4=15-36=-21. (n-1)a- (na-b)=b-a, and (n-2)a+b- (n−1)a=b-a, .. in formula for a put na-b, for n put fir t 8, and then 2n, and for ¿ put b-a; 1. Find the 7th, 10th, and 23rd terms of the series 1+3+5+7+&c. 2. Find the 6th, 11th, and 27th terms of the series 2+5+8+11+ &c. 3. Find the 4th, 7th, and (n-1)th terms of the series 11+8+5+ &c. 4. Find the 8th, 13th, and 2nth terms of the series 2+23+23+ &c. 6. Find the 5th, 12th, and (2n-1)th terms of the series 7. 1* ස Find the 12th, 20th, and nth terms of the series -5-3-1+&c. 8. Find the 7th, 19th, and (n+3)th terms of the series |