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If-the-sum-of-the-first-4-terms-of-an-A-P-is-p-the-sum-of-the-first-8-terms-is-q-and-the-sum-of-the-first-12-terms-is-r-express-3p-r-in-terms-of-q-




Question Number 11255 by 786786AM last updated on 18/Mar/17
If the sum of the first 4 terms of an A.P., is p, the sum of the first 8 terms is q and the sum of the first  12 terms is r, express (3p+r) in terms of q.
$$\mathrm{If}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{4}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{an}\:\mathrm{A}.\mathrm{P}.,\:\mathrm{is}\:\mathrm{p},\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first}\:\mathrm{8}\:\mathrm{terms}\:\mathrm{is}\:\mathrm{q}\:\mathrm{and}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{first} \\ $$$$\mathrm{12}\:\mathrm{terms}\:\mathrm{is}\:\mathrm{r},\:\mathrm{express}\:\left(\mathrm{3p}+\mathrm{r}\right)\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{q}. \\ $$
Answered by ajfour last updated on 18/Mar/17
p=2(2a+3d)  q=4(2a+7d)  r=6(2a+11d)  So, 3p+r=6(2a+3d)+6(2a+11d)                 = 6(4a+14d) =12(2a+7d)    3p+r =3q .
$$\mathrm{p}=\mathrm{2}\left(\mathrm{2a}+\mathrm{3d}\right) \\ $$$$\mathrm{q}=\mathrm{4}\left(\mathrm{2a}+\mathrm{7d}\right) \\ $$$$\mathrm{r}=\mathrm{6}\left(\mathrm{2a}+\mathrm{11d}\right) \\ $$$$\mathrm{So},\:\mathrm{3p}+\mathrm{r}=\mathrm{6}\left(\mathrm{2a}+\mathrm{3d}\right)+\mathrm{6}\left(\mathrm{2a}+\mathrm{11d}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{6}\left(\mathrm{4a}+\mathrm{14d}\right)\:=\mathrm{12}\left(\mathrm{2a}+\mathrm{7d}\right) \\ $$$$\:\:\mathrm{3p}+\mathrm{r}\:=\mathrm{3q}\:. \\ $$

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