Question Number 7449 by Tawakalitu. last updated on 29/Aug/16
$${The}\:{sum}\:{of}\:{the}\:{first}\:\mathrm{3}\:{terms}\:{of}\:{an}\:{AP}\:{is}\:\mathrm{36}\:{and}\:{the} \\ $$$${product}\:{of}\:{the}\:{first}\:\mathrm{3}\:{terms}\:{of}\:{a}\:{GP}\:{is}\:\mathrm{1728}.\:{if}\:{the} \\ $$$${second}\:{term}\:{of}\:{the}\:{linear}\:{sequence}\:{is}\:\mathrm{12}\:{and}\:{such} \\ $$$${that}\:{the}\:{second}\:{term}\:{of}\:{the}\:{GP}\:{is}\:{equal}\:{to}\:{the}\:{second} \\ $$$${term}\:{of}\:{the}\:{AP}.\:{find} \\ $$$$\left(\mathrm{1}\right)\:{The}\:{AP} \\ $$$$\left(\mathrm{2}\right)\:{The}\:{GP} \\ $$
Answered by Rasheed Soomro last updated on 30/Aug/16
![Let the common differnce of the AP is d As the second term is 12 The first term should be 12−d And the 3rd term 12+d The sum of the first three terms 36 The required AP is 12−d,12,12+d, 12+2d,... No matter what the value of d may be. Let the common ratio is r As the second term is also 12 So the first term should be 12/r and the 3rd term 12r The required GP is 12/r ,12,12r,12r^2 ,... No matter what the value of r may be Note: The sum of the three terms of AP [36] or GP [1728] has no role in determining the sequences.](Q7451.png)
$${Let}\:{the}\:{common}\:{differnce}\:{of}\:{the}\:{AP}\:{is}\:\:{d} \\ $$$${As}\:{the}\:{second}\:{term}\:{is}\:\:\mathrm{12} \\ $$$${The}\:{first}\:{term}\:{should}\:{be}\:\mathrm{12}−{d} \\ $$$${And}\:{the}\:\mathrm{3}{rd}\:{term}\:\mathrm{12}+{d} \\ $$$${The}\:{sum}\:{of}\:{the}\:{first}\:{three}\:{terms}\:\:\mathrm{36} \\ $$$${The}\:{required}\:{AP}\:{is} \\ $$$$\mathrm{12}−{d},\mathrm{12},\mathrm{12}+{d},\:\mathrm{12}+\mathrm{2}{d},...\:{No}\:{matter}\:{what}\:{the} \\ $$$${value}\:{of}\:{d}\:{may}\:{be}. \\ $$$$\:\:\:\: \\ $$$${Let}\:{the}\:{common}\:{ratio}\:{is}\:\:{r} \\ $$$${As}\:{the}\:{second}\:{term}\:{is}\:{also}\:\mathrm{12} \\ $$$${So}\:{the}\:{first}\:{term}\:{should}\:{be}\:\mathrm{12}/{r} \\ $$$${and}\:{the}\:\mathrm{3}{rd}\:{term}\:\:\:\:\:\:\:\:\:\:\:\mathrm{12}{r} \\ $$$${The}\:{required}\:{GP}\:{is} \\ $$$$\mathrm{12}/{r}\:,\mathrm{12},\mathrm{12}{r},\mathrm{12}{r}^{\mathrm{2}} ,...\:{No}\:{matter}\:{what}\:{the}\:{value} \\ $$$${of}\:{r}\:\:{may}\:{be} \\ $$$${Note}:\:{The}\:{sum}\:{of}\:{the}\:\:{three}\:{terms}\:{of}\:{AP}\:\:\left[\mathrm{36}\right] \\ $$$${or}\:{GP}\:\left[\mathrm{1728}\right]\:{has}\:{no}\:{role}\:{in}\:{determining}\:{the}\:{sequences}. \\ $$$$ \\ $$
Commented by Rasheed Soomro last updated on 30/Aug/16
$${There}\:{are}\:{many}\:{AP}\:'{s}\:{and}\:{GP}\:'{s}\:,\:{which}\:{fulfil} \\ $$$${the}\:{conditions}\:{given}\:{above}. \\ $$$$``{The}\:{sum}\:{of}\:{the}\:{terms}\:{of}\:\:{AP}\:\:{or}\:\:{GP}\:\:{to}\:{be}\:\mathrm{36}\:{and}\:\mathrm{1728}\:{respectively}'', \\ $$$${the}\:{condition}\:{given}\:{in}\:{the}\:{question}\:{is}\:\boldsymbol{{unnecessary}}. \\ $$
Commented by Tawakalitu. last updated on 31/Aug/16
$${Thank}\:{you}\:{so}\:{much}\:{sir} \\ $$
Answered by Rasheed Soomro last updated on 01/Sep/16
$${Different}\:{Approach} \\ $$$${Let}\:{the}\:{common}\:{difference}\:{is}\:{d}\:{and}\:{second}\:{term}\:{is}\:\:\:{a} \\ $$$${The}\:{first}\:{three}\:{terms}\:{of}\:{the}\:{AP}\:\:{will}\:{be}\::\:\:\:{a}−{d},{a},{a}+{d} \\ $$$${Sum}\:{is}\:\mathrm{36}\::\:\left({a}−{d}\right)+{a}+\left({a}+{d}\right)=\mathrm{36} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}{a}=\mathrm{36} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}=\mathrm{12} \\ $$$${The}\:{AP}:\:\mathrm{12}−{d}\:,\:\mathrm{12},\mathrm{12}+{d},\mathrm{12}+\mathrm{2}{d},.... \\ $$$${Let}\:{the}\:{common}\:{ratio}\:{is}\:{d}\:{and}\:{second}\:{term}\:{is}\:\:\:{b} \\ $$$${The}\:{first}\:{three}\:{terms}\:{of}\:{the}\:{GP}\:{will}\:{be}\:{b}/{r}\:,{b},\:{br} \\ $$$${The}\:{product}\:{is}\:\mathrm{1728}:\:\frac{{b}}{{r}}×{b}×{br}=\mathrm{1728} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{b}^{\mathrm{3}} =\mathrm{1728} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{b}=\mathrm{12} \\ $$$${The}\:{GP}:\:\:\mathrm{12}/{r}\:,\:\mathrm{12},\:\mathrm{12}{r}\:,\mathrm{12}{r}^{\mathrm{2}} ,... \\ $$$${Note}:\:{The}\:{second}\:{term}\:{of}\:{the}\:{both},\:{AP}\:\:{and}\:{GP}\:{is}\:{given} \\ $$$${but}\:{in}\:{this}\:{approach}\:{that}\:{information}\:{has}\:{not}\:{been}\:{used} \\ $$$${deliberately}.{Because}\:{any}\:{one}\:{of}\:{the}\:{following}\:{informations} \\ $$$${is}\:\:\boldsymbol{{sufficient}}\:{in}\:{determining}\:{the}\:{AP}\:{and}\:{GP}\:\:{and}\:{the}\:{other} \\ $$$${is}\:\boldsymbol{{unnecessary}}. \\ $$$$\left({i}\right)\:\mathrm{2}{nd}\:{term}\:{of}\:{both}\:{AP}\:{and}\:{GP}\:{is}\:\mathrm{12} \\ $$$$\left({ii}\right)\:{The}\:{sum}\:{of}\:{the}\:{first}\:{three}\:{terms}\:{of}\:{the}\:{AP}\:\:\:{is}\:\:\mathrm{36} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{and}\:{the}\:{product}\:{of}\:{the}\:{GP}\:\:{is}\:\mathrm{1728}. \\ $$$$\:\:\:\:\: \\ $$