By K. F. Riley, M. P. Hobson
This scholar answer handbook offers whole strategies to all of the odd-numbered difficulties in beginning arithmetic for the actual Sciences. It takes scholars via each one challenge step by step, to allow them to essentially see how the answer is reached, and comprehend any error of their personal operating. scholars will examine via instance the best way to arrive on the right solution and increase their problem-solving abilities.
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Extra resources for Foundation Mathematics for the Physical Sciences: Student Solution Manual
This is the same form as in the assumption except that N has been replaced by N + 1; this shows that the result is valid for n = N + 1 if it is valid for n = N. But the assumed result is trivially valid for n = 1 and is therefore valid for all n. To prove that n r 3 = 14 n2 (n + 1)2 , r=1 assume that the result is valid for n = N and consider N+1 N r = r 3 + (N + 1)3 3 r=1 r=1 = 14 N 2 (N + 1)2 + (N + 1)3 , using the assumption, = 14 (N + 1)2 [ N 2 + 4(N + 1) ] = 14 (N + 1)2 (N + 2)2 . This is the same form as in the assumption except that N has been replaced by N + 1 and shows that the result is valid for n = N + 1 if it is valid for n = N.
3x − 10 x+5 x−2 (i) A B x(A + B) + (5B − 2A) + = . x+5 x−2 (x + 5)(x − 2) Solving A + B = 2 and −2A + 5B = 1 gives A = 97 and B = 57 . (ii) Setting x equal to 0 and 1, say, gives the pair of equations 1 A B = + ; −10 5 −2 −1 = 2A − 5B; with solution A = 9 7 and B = 57 . 3 A B = + , −6 6 −1 −3 = A − 6B, 23 Preliminary algebra (iii) A= 2(−5) + 1 9 = ; −5 − 2 7 B= 2(2) + 1 5 = . 2+5 7 All three methods give the same decomposition. (b) Here the factorisation of the denominator is simply x(x − 3) or, more formally, (x − 0)(x − 3), and the expansion takes the form 4 A B = + .
N2 − 1 50 Integral calculus Taking sin nθ as u and cos θ as v and noting that with this choice u = −n2 u and v = −v, we expect that after two integrations by parts we will recover (a multiple of) In . π/2 In = sin nθ cos θ dθ 0 π/2 = [sin nθ sin θ ]0 π/2 − n cos nθ sin θ dθ 0 π/2 nπ π/2 − n [− cos nθ cos θ]0 − 2 nπ = sin − n[ −(−1) − nIn ]. 2 = sin (−n sin nθ)(− cos θ ) dθ 0 Rearranging this gives In (1 − n2 ) = sin nπ − n, 2 and hence the stated result. )/2, (b) J2r = 2−r (2r − 1)(2r − 3) · · · (5)(3)(1) J0 .