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What-is-the-next-term-in-the-below-sequence-1-3-9-27-81-243-729-2123-5857-




Question Number 3105 by prakash jain last updated on 04/Dec/15
What is the next term in the below sequence  1,3,9,27,81,243,729, 2123, 5857,?
$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{next}\:\mathrm{term}\:\mathrm{in}\:\mathrm{the}\:\mathrm{below}\:\mathrm{sequence} \\ $$$$\mathrm{1},\mathrm{3},\mathrm{9},\mathrm{27},\mathrm{81},\mathrm{243},\mathrm{729},\:\mathrm{2123},\:\mathrm{5857},? \\ $$
Commented by Filup last updated on 04/Dec/15
3^(10) ?
$$\mathrm{3}^{\mathrm{10}} ? \\ $$
Commented by prakash jain last updated on 05/Dec/15
2123≠3^7
$$\mathrm{2123}\neq\mathrm{3}^{\mathrm{7}} \\ $$
Commented by Filup last updated on 05/Dec/15
 ^� �\(^• −^• )/ ^�  ^�   You have me stumped on this one.  For now...
$$\bar {\:}\backslash\left(\:^{\bullet} −^{\bullet} \right)/\bar {\:}\bar {\:} \\ $$$$\mathrm{You}\:\mathrm{have}\:\mathrm{me}\:\mathrm{stumped}\:\mathrm{on}\:\mathrm{this}\:\mathrm{one}. \\ $$$$\mathrm{For}\:\mathrm{now}… \\ $$
Commented by prakash jain last updated on 05/Dec/15
I was trying to highlight a point that given  a finite number of terms there are infinite  number of formula that can be defined to  generate the next term.  So if you were given only first 7 term the   obvious formula is 3^(n−1) , while there may  be other formulas as well.
$$\mathrm{I}\:\mathrm{was}\:\mathrm{trying}\:\mathrm{to}\:\mathrm{highlight}\:\mathrm{a}\:\mathrm{point}\:\mathrm{that}\:\mathrm{given} \\ $$$$\mathrm{a}\:\mathrm{finite}\:\mathrm{number}\:\mathrm{of}\:\mathrm{terms}\:\mathrm{there}\:\mathrm{are}\:\mathrm{infinite} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{formula}\:\mathrm{that}\:\mathrm{can}\:\mathrm{be}\:\mathrm{defined}\:\mathrm{to} \\ $$$$\mathrm{generate}\:\mathrm{the}\:\mathrm{next}\:\mathrm{term}. \\ $$$$\mathrm{So}\:\mathrm{if}\:\mathrm{you}\:\mathrm{were}\:\mathrm{given}\:\mathrm{only}\:\mathrm{first}\:\mathrm{7}\:\mathrm{term}\:\mathrm{the}\: \\ $$$$\mathrm{obvious}\:\mathrm{formula}\:\mathrm{is}\:\mathrm{3}^{{n}−\mathrm{1}} ,\:\mathrm{while}\:\mathrm{there}\:\mathrm{may} \\ $$$$\mathrm{be}\:\mathrm{other}\:\mathrm{formulas}\:\mathrm{as}\:\mathrm{well}. \\ $$
Commented by Rasheed Soomro last updated on 05/Dec/15
You have answered my mind ′s question before expressing!!
$${You}\:{have}\:{answered}\:{my}\:{mind}\:'{s}\:{question}\:{before}\:{expressing}!! \\ $$
Commented by Rasheed Soomro last updated on 05/Dec/15
    15203
$$\:\:\:\:\mathrm{15203} \\ $$
Answered by Rasheed Soomro last updated on 06/Dec/15
1         3       9       27      81       243        729          2123         5857      15203_(−) ^((5857−(−9346))        −2       −6   −18  −54  −162  −486   −1394     −3734                     −9346^((−3734−5612))                4       12       36       108      324      908          2340                 5612^((2340−(−3272))                −8    −24    −72     −216  −584   −1432            −3272^((−1432−1840))                       16       48         144        368      848                 1840^((848−(−992)))                            −32     −96     −224   −480          −992^((−480−512))                                     64        128         256         512^((2^9 ))   The lines written in black are the given sequence   and derived sequences. Each sequenc is made by  subtraction of consecutive terms of previous  sequence.  Last sequence is GP. Its next term is 512.  Previous to the last is calculated as −992  Calculating next terms in reverse order,  The next term of the given sequence is     15203
$$\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{3}\:\:\:\:\:\:\:\mathrm{9}\:\:\:\:\:\:\:\mathrm{27}\:\:\:\:\:\:\mathrm{81}\:\:\:\:\:\:\:\mathrm{243}\:\:\:\:\:\:\:\:\mathrm{729}\:\:\:\:\:\:\:\:\:\:\mathrm{2123}\:\:\:\:\:\:\:\:\:\mathrm{5857}\:\:\:\:\:\:\underset{−} {\overset{\left(\mathrm{5857}−\left(−\mathrm{9346}\right)\right.} {\mathrm{15203}}}\:\:\: \\ $$$$\:\:−\mathrm{2}\:\:\:\:\:\:\:−\mathrm{6}\:\:\:−\mathrm{18}\:\:−\mathrm{54}\:\:−\mathrm{162}\:\:−\mathrm{486}\:\:\:−\mathrm{1394}\:\:\:\:\:−\mathrm{3734}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overset{\left(−\mathrm{3734}−\mathrm{5612}\right)} {−\mathrm{9346}}\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{4}\:\:\:\:\:\:\:\mathrm{12}\:\:\:\:\:\:\:\mathrm{36}\:\:\:\:\:\:\:\mathrm{108}\:\:\:\:\:\:\mathrm{324}\:\:\:\:\:\:\mathrm{908}\:\:\:\:\:\:\:\:\:\:\mathrm{2340}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overset{\left(\mathrm{2340}−\left(−\mathrm{3272}\right)\right.} {\mathrm{5612}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:−\mathrm{8}\:\:\:\:−\mathrm{24}\:\:\:\:−\mathrm{72}\:\:\:\:\:−\mathrm{216}\:\:−\mathrm{584}\:\:\:−\mathrm{1432}\:\:\:\:\:\:\:\:\:\:\:\:\overset{\left(−\mathrm{1432}−\mathrm{1840}\right)} {−\mathrm{3272}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{16}\:\:\:\:\:\:\:\mathrm{48}\:\:\:\:\:\:\:\:\:\mathrm{144}\:\:\:\:\:\:\:\:\mathrm{368}\:\:\:\:\:\:\mathrm{848}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overset{\left(\mathrm{848}−\left(−\mathrm{992}\right)\right)} {\mathrm{1840}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\mathrm{32}\:\:\:\:\:−\mathrm{96}\:\:\:\:\:−\mathrm{224}\:\:\:−\mathrm{480}\:\:\:\:\:\:\:\:\:\:\overset{\left(−\mathrm{480}−\mathrm{512}\right)} {−\mathrm{992}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{64}\:\:\:\:\:\:\:\:\mathrm{128}\:\:\:\:\:\:\:\:\:\mathrm{256}\:\:\:\:\:\:\:\:\:\overset{\left(\mathrm{2}^{\mathrm{9}} \right)} {\mathrm{512}} \\ $$$$\mathcal{T}{he}\:{lines}\:{written}\:{in}\:{black}\:{are}\:{the}\:{given}\:{sequence}\: \\ $$$${and}\:{derived}\:{sequences}.\:\mathcal{E}{ach}\:{sequenc}\:{is}\:{made}\:{by} \\ $$$${subtraction}\:{of}\:{consecutive}\:{terms}\:{of}\:{previous} \\ $$$${sequence}. \\ $$$$\mathcal{L}{ast}\:{sequence}\:{is}\:{GP}.\:{Its}\:{next}\:{term}\:{is}\:\mathrm{512}. \\ $$$$\mathcal{P}{revious}\:{to}\:{the}\:{last}\:{is}\:{calculated}\:{as}\:−\mathrm{992} \\ $$$$\mathcal{C}{alculating}\:{next}\:{terms}\:{in}\:{reverse}\:{order}, \\ $$$$\mathcal{T}{he}\:{next}\:{term}\:{of}\:{the}\:{given}\:{sequence}\:{is}\:\:\:\:\:\mathrm{15203} \\ $$
Commented by Rasheed Soomro last updated on 05/Dec/15
The above answer could be possible only in light of your  (prakash′s ) guidance!
$$\mathcal{T}{he}\:{above}\:{answer}\:{could}\:{be}\:{possible}\:{only}\:{in}\:{light}\:{of}\:{your} \\ $$$$\left({prakash}'{s}\:\right)\:{guidance}! \\ $$
Commented by prakash jain last updated on 06/Dec/15
Successive difference techinque is based on  the fact that it reduces degree of polynomials  and exponents by 1.
$$\mathrm{Successive}\:\mathrm{difference}\:\mathrm{techinque}\:\mathrm{is}\:\mathrm{based}\:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{fact}\:\mathrm{that}\:\mathrm{it}\:\mathrm{reduces}\:\mathrm{degree}\:\mathrm{of}\:\mathrm{polynomials} \\ $$$$\mathrm{and}\:\mathrm{exponents}\:\mathrm{by}\:\mathrm{1}.\: \\ $$

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