# If-the-G-M-between-two-numbers-a-b-is-G-and-the-two-A-M-s-between-them-are-p-and-q-then-prove-that-G-2-2p-q-2q-p-

Question Number 3617 by Rasheed Soomro last updated on 16/Dec/15
$${If}\:\:{the}\:{G}.{M}.\:{between}\:{two}\:{numbers}\: \\$$$${a}\:,\:{b}\:\:\:{is}\:\:{G}\:{and}\:{the}\:{two}\:{A}.{M}.'{s}\:{between} \\$$$${them}\:{are}\:{p}\:\:{and}\:\:{q}\:,\:{then}\:{prove}\:{that} \\$$$${G}^{\mathrm{2}} =\left(\mathrm{2}{p}−{q}\right)\left(\mathrm{2}{q}−{p}\right). \\$$
Commented by Yozzii last updated on 16/Dec/15
$${There}\:{are}\:{multiple}\:{definitions}\:{of}\:{the} \\$$$${A}.{M}\:{between}\:{a}\:{and}\:{b}? \\$$
Commented by prakash jain last updated on 16/Dec/15
$$\mathrm{AP}:\:{a},\:{a}+{d},\:{a}+\mathrm{2}{d},\:{b}={a}+\mathrm{3}{d} \\$$$${p}={a}+{d},{q}={a}+\mathrm{2}{d} \\$$$$\mathrm{2}{p}−{q}=\mathrm{2}{a}+\mathrm{2}{d}−{a}+\mathrm{2}{d}={a} \\$$$$\mathrm{2}{q}−{p}=\mathrm{2}\left({a}+\mathrm{2}{d}\right)−\left({a}+{d}\right)={a}+\mathrm{3}{d}={b} \\$$$$\left(\mathrm{2}{p}−{q}\right)\left(\mathrm{2}{q}−{p}\right)={ab}={G}^{\mathrm{2}} \\$$
Commented by prakash jain last updated on 16/Dec/15
$${n}\:\mathrm{AM}{s}\:{x}_{{i}} \:\mathrm{between}\:{a}\:\mathrm{and}\:{b}\:\mathrm{means} \\$$$${a},{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,…,{x}_{{n}} ,{b}\:\mathrm{is}\:\mathrm{an}\:\mathrm{AP}. \\$$