Question Number 11930 by 786786AM last updated on 05/Apr/17
$$\mathrm{Let}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{be}\:\mathrm{two}\:\mathrm{numbers},\:\mathrm{x}\:\mathrm{be}\:\mathrm{the}\:\mathrm{single}\:\mathrm{arithmetic}\:\mathrm{mean}\:\mathrm{of}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}. \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{n}\:\mathrm{arithmetic}\:\mathrm{means}\:\mathrm{between}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{is}\:\mathrm{nx}. \\ $$
Answered by ajfour last updated on 05/Apr/17
$${A}_{{r}} =\:{a}+\frac{\left({b}−{a}\right)}{\left({n}+\mathrm{1}\right)}{r} \\ $$$$\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{A}_{{r}} \:=\:{na}+\frac{\left({b}−{a}\right)}{\left({n}+\mathrm{1}\right)}.\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:={na}+\frac{{n}\left({b}−{a}\right)}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{{n}\left({a}+{b}\right)}{\mathrm{2}}\:=\:{nx}\:. \\ $$