Question Number 46923 by 786786AM last updated on 02/Nov/18
$${a},\:{b},\:{c}\:\mathrm{are}\:\mathrm{in}\:\mathrm{AP}\:\mathrm{or}\:\mathrm{GP}\:\mathrm{or}\:\mathrm{HP}\:\mathrm{according} \\ $$$$\mathrm{as}\:\frac{{a}−{b}}{{b}−{c}}\:\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 02/Nov/18
$$ \\ $$$$\left.\mathrm{1}\right)\:{when}\:{a},{b},{c}\:{are}\:{in}\:{A}.{P} \\ $$$${then}\:{a}−{b}={b}−{c}\:\:\:\:{so}\:\frac{{a}−{b}}{{b}−{c}}=\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{when}\:{a},{b},{c}\:{are}\:{in}\:{GP} \\ $$$${b}={ar}\:\:\:{c}={ar}^{\mathrm{2}} \\ $$$$\frac{{a}−{b}}{{b}−{c}}=\frac{{a}−{ar}}{{ar}−{ar}^{\mathrm{2}} }=\frac{{a}\left(\mathrm{1}−{r}\right)}{{ar}\left(\mathrm{1}−{r}\right)}=\frac{\mathrm{1}}{{r}}={reciprocal}\:{of}\: \\ $$$${common}\:{ratio} \\ $$$$\left.\mathrm{3}\right){when}\:{a},{b},{c}\:{are}\:{in}\:{GP} \\ $$$$\frac{\mathrm{2}}{{b}}=\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{c}} \\ $$$$\frac{\mathrm{1}}{{b}}−\frac{\mathrm{1}}{{a}}=\frac{\mathrm{1}}{{c}}−\frac{\mathrm{1}}{{b}} \\ $$$$\frac{{a}−{b}}{{ab}}=\frac{{b}−{c}}{{bc}} \\ $$$$\frac{{a}−{b}}{{b}−{c}}=\frac{{a}}{{c}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$