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The-sum-of-the-first-3-terms-of-an-AP-is-36-and-the-product-of-the-first-3-terms-of-a-GP-is-1728-if-the-second-term-of-the-linear-sequence-is-12-and-such-that-the-second-term-of-the-GP-is-equal-to-th




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

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