Download Professor Higgins’s Problem Collection by Peter M. Higgins PDF
By Peter M. Higgins
What are you able to do together with your maths? you should use it to entirely comprehend all demeanour of items that can't be handled in the other manner. This publication serves up a number of difficulties and exhibits how arithmetic solutions them. subject matters diversity from cracking codes to the patience of recessive genes; from common sense puzzles to classical geometry; and from planetary movement inquiries to predicting the industry proportion of competing businesses. And there are different difficulties the place the maths itself is intrinsically unbelievable and interesting.
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Extra resources for Professor Higgins’s Problem Collection
Sample text
Hence we arrive at n (n + 1)2 – 12 = 2 n m+ m=1 1 m=1 n ⇒ n2 + 2n + 1 – 1 = 2 m +n m=1 n ⇒ m= m=1 (b) n2 + n 1 = n(n + 1). 2 2 We may use the same approach applied to the difference of consecutive cubes, along with the formula for the sum of one-powers that we have just derived, in order to find the sum of consecutive squares: n n [(m + 1)3 – m3 ] = m=1 (3m2 + 3m + 1) m=1 n ⇒ (n + 1)3 – 13 = 3 n m2 + 3 m=1 n ⇒ (n + 1)3 – 13 = 3 n m+ m=1 1 m=1 3 m2 + n(n + 1) + n 2 m=1 n ⇒3 3 3 m2 = n3 + 3n2 + 3n – n2 – n – n 2 2 m=1 n ⇒ m2 = m=1 n ∴ m2 = m=1 1 3 3 2 n n + n + 3 2 2 n n 2n2 + 3n + 1 = (n + 1)(2n + 1).
C + di c – di c2 + d2 c + d2 c + d2 ( 6) It is not important, however, to memorize either formulae (5) or (6), for rather we just need to know how to carry out the calculation in any particular case. A complex number z = x + iy may be written in polar form as z = reiθ = r(cos θ + i sin θ), where r = x2 + y2 and tan θ = yx ; the number r ≥ 0 is the distance of the point P(x, y) in the plane from the origin, while θ is the angle between the real axis (the x-axis) and the ray from the origin to P.
What number is this and what equation are you thereby solving? y 1 y=x ? y = cos x –π – π 2 π 2 0 –1 π x 44 PROFESSOR HIGGINS’ S PROBLEM COLLECTION Problem 18 solution (a) The quickest way to do this is first to multiply by sin 20◦ and then keep applying the double angle formula, sin x cos x = 12 sin 2x. With this in mind, put a = cos 20◦ cos 40◦ cos 80◦ . Then, following the advice above, we obtain 1 sin 40◦ cos 40◦ cos 80◦ 2 1 1 1 = sin 80◦ cos 80◦ = sin 160◦ = sin 20◦ , 4 8 8 a sin 20◦ = sin 20◦ cos 20◦ cos 40◦ cos 80◦ = where the final equality is justified by the general relationship, sin(180 – x)◦ = sin x◦ .