psyclones Posted February 15, 2015 Share Posted February 15, 2015 (edited) Hi All,I found this problem, The sum of p, q, r terms of an Arithmetic Progression, are P, Q, R respectively: show that[latex] \frac{P (q - r )}{p} + \frac{Q (r - p )}{q} + \frac{R (p - q)}{r} = 0 [/latex] My thoughts on how to start the problem is; if[latex] S_{n} = \frac{a}{2} (n + (n-1)d ) [/latex]then the sum of say 'p' terms, would be[latex] P = S_{p} = \frac{a}{2} (p + (p-1)d ) [/latex]Therefore;[latex] Q = S_{q} = \frac{a}{2} (q + (q-1)d ) [/latex][latex] R = S_{r} = \frac{a}{2} (r + (r-1)d ) [/latex]If I used the following series, to simplify the above P, Q & R series; [latex] S_{n} = 1 + 2 + 3 ... + n, [/latex] then [latex] S_{n} = \frac{1}{2}n(n+1) [/latex]But how to form the above equation, which contains all the terms (i.e, p, q, r, P, Q & R)? Edited February 15, 2015 by psyclones Link to comment Share on other sites More sharing options...
imatfaal Posted February 15, 2015 Share Posted February 15, 2015 S_n = n/2(2a_0+(n-1)d) I think you have your n and a_0 switched at first glance yes - it works You have an equation in P Q R and p q r - and you can set up P in terms of p a and d etc. If you correctly set up the sum equation - then sub in three times with expressions in a, d, p,r, and q. Simplify. It works If you struggle I have done it and have a hand-written confirmation - it's a bit too lengthy to change into latex. So I cannot give you a line at a time - just paste the whole thing as a jpg. Link to comment Share on other sites More sharing options...
psyclones Posted February 17, 2015 Author Share Posted February 17, 2015 Thank-you for your post, I solved it![latex] \frac{P}{p} = \frac{1}{2}(a + d(p-1)) [/latex] [latex] (eq 1) [/latex]Use sum of Q, to let a be the subject.[latex] a = 2\frac{Q}{q} - d(q-1) [/latex] [latex] (eq 2) [/latex]Sub, eq 1 into eq 2[latex] d = 2 \frac{Q}{(p-q)} ( \frac{P}{p} - \frac{Q}{q} ) [/latex]Do the same for Sum Q & R, solve for d.[latex] d = 2 \frac{Q}{(p-r)} ( \frac{Q}{q} - \frac{R}{r} ) [/latex][latex] 2 \frac{Q}{(p-q)} ( \frac{P}{p} - \frac{Q}{q} ) = 2 \frac{Q}{(p-r)} ( \frac{Q}{q} - \frac{R}{r} ) [/latex]Simplify,[latex] \frac{P (q - r )}{p} + \frac{Q (r - p )}{q} + \frac{R (p - q)}{r} = 0 [/latex] Link to comment Share on other sites More sharing options...
imatfaal Posted February 17, 2015 Share Posted February 17, 2015 Link to comment Share on other sites More sharing options...
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