# A Manual of Practical Mathematics

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1908 edition. Excerpt: ...quantities a and c. Then-A=a-, Q=Ja7, H = 2 a + c EXERCISES. XXXI. 1. Define harmonic progression; insert 4 harmonic means between 2 and 12. 2. Find the arithmetic, geometric, and harmonic means between 2 and 8. 3. Find a third term to 42 and 12 in H.p. 4. Find a first term to 8 and 20 in H.p. 5. The sum of three terms in H.p. is l-jV; if the first term is, what is the series? 6. The arithmetical mean between two numbers exceeds the geometric by 2, and the geometrical exceeds the harmonical by 1-6. Find the numbers. 7. A H.p. consists of six terms; the last three terms are 2, 3 and 6; find the first three. 8. Find in h.p. the fourth term to 6, 8 and 12. 9. Insert three harmonic means between 2 and 3. 10. Find the arithmetic, geometric, and harmonic means between 2 and f-, and write down three terms of each series. 11. If x, y, z be the ptb, qtb and rib terms of a H.p., show that (r-q) yz + (p-r)xz + (q-p) xy = 0. Miscellaneous Series.--The preceding methods may sometimes be adopted to obtain the summation of given series neither in A.p. nor in G.p. The processes employed may be seen from the following examples: Ex. 1. (a) Find the sum of the series a + (a + b)x + (a + 2b)x2 +... + a+ (n-l)bx"-1. (6) Show that the sum of the first n even numbers is equal to times the sum of the first n odd numbers.: . sum of first n even numbers = ( 1 +-I times the sum of the 9. 2 + f+ ff, etc. 10. What is meant by the sum of a geometrical series to infinity? Given that r is positive and that (l+r+?a+... to infinity)(1 +p+p1 +... to infinity) = 1 + rp + rp2 +..., show that p must be negative and r less than 11. The first and second terms of a progression are 5j and 2Find the 4th term on the supposition that the progression is (o) A.p., (6) g.p., ...show more

## Product details

• Paperback | 96 pages
• 189 x 246 x 5mm | 186g
• Miami Fl, United States
• English
• black & white illustrations
• 1236553535
• 9781236553539