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Determine-nth-term-of-the-following-sequence-1-0-1-0-7-28-79-




Question Number 3061 by Rasheed Soomro last updated on 04/Dec/15
Determine  nth  term of the following sequence  1,0,−1,0,7,28,79,...
$$\mathrm{Determine}\:\:\mathrm{nth}\:\:\mathrm{term}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{sequence} \\ $$$$\mathrm{1},\mathrm{0},−\mathrm{1},\mathrm{0},\mathrm{7},\mathrm{28},\mathrm{79},… \\ $$
Commented by prakash jain last updated on 04/Dec/15
one techinque is to define  define a series by successive subtraction  until you get a known pattern.  a_1 =1  a_2 =0  a_3 =−1  a_4 =0  a_5 =7  a_6 =28  a_7 =79  b_2 =a_2 −a_1 =−1  b_3 =−1  b_4 =1, b_5 =7, b_6 =21,b_7 =51  similary c_n =b_n −b_(n−1)   c_3 =0  c_4 =2 c_5 =6 c_6 =14 c_7 =30  d_n =c_n −c_(n−1)   d_4 =2 d_5 =4 d_6 =8 d_7 =16  d_n =2^(n−3)   (n≥4)  c_n =c_(n−1) +d_n (n≥4)  b_n =b_(n−1) +c_n (n≥3)  a_n =a_(n−1) +b_n (n≥2)  for 8^(th)  tem  d_8 =32  c_8 =c_7 +d_8 =30+32=62  b_8 =b_7 +c_8 =51+62=113  a_8 =a_7 +b_8 =79+113=192
$${one}\:{techinque}\:{is}\:{to}\:{define} \\ $$$${define}\:{a}\:{series}\:{by}\:{successive}\:{subtraction} \\ $$$${until}\:{you}\:{get}\:{a}\:{known}\:{pattern}. \\ $$$${a}_{\mathrm{1}} =\mathrm{1} \\ $$$${a}_{\mathrm{2}} =\mathrm{0} \\ $$$${a}_{\mathrm{3}} =−\mathrm{1} \\ $$$${a}_{\mathrm{4}} =\mathrm{0} \\ $$$${a}_{\mathrm{5}} =\mathrm{7} \\ $$$${a}_{\mathrm{6}} =\mathrm{28} \\ $$$${a}_{\mathrm{7}} =\mathrm{79} \\ $$$${b}_{\mathrm{2}} ={a}_{\mathrm{2}} −{a}_{\mathrm{1}} =−\mathrm{1} \\ $$$${b}_{\mathrm{3}} =−\mathrm{1} \\ $$$${b}_{\mathrm{4}} =\mathrm{1},\:{b}_{\mathrm{5}} =\mathrm{7},\:{b}_{\mathrm{6}} =\mathrm{21},{b}_{\mathrm{7}} =\mathrm{51} \\ $$$${similary}\:{c}_{{n}} ={b}_{{n}} −{b}_{{n}−\mathrm{1}} \\ $$$${c}_{\mathrm{3}} =\mathrm{0}\:\:{c}_{\mathrm{4}} =\mathrm{2}\:{c}_{\mathrm{5}} =\mathrm{6}\:{c}_{\mathrm{6}} =\mathrm{14}\:{c}_{\mathrm{7}} =\mathrm{30} \\ $$$${d}_{{n}} ={c}_{{n}} −{c}_{{n}−\mathrm{1}} \\ $$$${d}_{\mathrm{4}} =\mathrm{2}\:{d}_{\mathrm{5}} =\mathrm{4}\:{d}_{\mathrm{6}} =\mathrm{8}\:{d}_{\mathrm{7}} =\mathrm{16} \\ $$$${d}_{{n}} =\mathrm{2}^{{n}−\mathrm{3}} \:\:\left({n}\geqslant\mathrm{4}\right) \\ $$$${c}_{{n}} ={c}_{{n}−\mathrm{1}} +{d}_{{n}} \left({n}\geqslant\mathrm{4}\right) \\ $$$${b}_{{n}} ={b}_{{n}−\mathrm{1}} +{c}_{{n}} \left({n}\geqslant\mathrm{3}\right) \\ $$$${a}_{{n}} ={a}_{{n}−\mathrm{1}} +{b}_{{n}} \left({n}\geqslant\mathrm{2}\right) \\ $$$${for}\:\mathrm{8}^{{th}} \:{tem} \\ $$$${d}_{\mathrm{8}} =\mathrm{32} \\ $$$${c}_{\mathrm{8}} ={c}_{\mathrm{7}} +{d}_{\mathrm{8}} =\mathrm{30}+\mathrm{32}=\mathrm{62} \\ $$$${b}_{\mathrm{8}} ={b}_{\mathrm{7}} +{c}_{\mathrm{8}} =\mathrm{51}+\mathrm{62}=\mathrm{113} \\ $$$${a}_{\mathrm{8}} ={a}_{\mathrm{7}} +{b}_{\mathrm{8}} =\mathrm{79}+\mathrm{113}=\mathrm{192} \\ $$
Commented by Rasheed Soomro last updated on 05/Dec/15
THankS  ^(for)    Guidance^(Valueable)  !
$$\mathcal{TH}{ank}\mathcal{S}\:\:\:^{{for}} \:\:\:\overset{\mathcal{V}{alueable}} {\mathcal{G}{uidance}}\:! \\ $$
Answered by Filup last updated on 04/Dec/15
1=2−1  ⇒  2^1 −1  0=4−4  ⇒  2^2 −2^2   −1=8−9  ⇒  2^3 −3^2   etc.    T_n =2^n −n^2
$$\mathrm{1}=\mathrm{2}−\mathrm{1}\:\:\Rightarrow\:\:\mathrm{2}^{\mathrm{1}} −\mathrm{1} \\ $$$$\mathrm{0}=\mathrm{4}−\mathrm{4}\:\:\Rightarrow\:\:\mathrm{2}^{\mathrm{2}} −\mathrm{2}^{\mathrm{2}} \\ $$$$−\mathrm{1}=\mathrm{8}−\mathrm{9}\:\:\Rightarrow\:\:\mathrm{2}^{\mathrm{3}} −\mathrm{3}^{\mathrm{2}} \\ $$$${etc}. \\ $$$$ \\ $$$${T}_{{n}} =\mathrm{2}^{{n}} −{n}^{\mathrm{2}} \\ $$
Commented by Rasheed Soomro last updated on 04/Dec/15
V^(ery ) G^(OO) D!
$$\mathcal{V}\:^{{ery}\:} \mathcal{G}^{\mathcal{OO}} \mathcal{D}! \\ $$
Commented by RasheedAhmad last updated on 04/Dec/15
  How did you reach that solution!  Is there a way? I think these types  of questions like ′ riddles ′ which  can be solved only by guesses!   Although intelligent guesses!
$$ \\ $$$$\mathcal{H}{ow}\:{did}\:{you}\:{reach}\:{that}\:{solution}! \\ $$$${Is}\:{there}\:{a}\:{way}?\:{I}\:{think}\:{these}\:{types} \\ $$$${of}\:{questions}\:{like}\:'\:{riddles}\:'\:{which} \\ $$$${can}\:{be}\:{solved}\:{only}\:{by}\:{guesses}!\: \\ $$$${Although}\:{intelligent}\:{guesses}! \\ $$
Commented by prakash jain last updated on 04/Dec/15
See comments in question about approach  to find general terms. Uniqueness of  formula is not guaranteed when only finite  number of terms are known.  Also see quesion 3003, where two different  formulas can generate all known terms  correctly but differ on next term.
$$\mathrm{See}\:\mathrm{comments}\:\mathrm{in}\:\mathrm{question}\:\mathrm{about}\:\mathrm{approach} \\ $$$$\mathrm{to}\:\mathrm{find}\:\mathrm{general}\:\mathrm{terms}.\:\mathrm{Uniqueness}\:\mathrm{of} \\ $$$$\mathrm{formula}\:\mathrm{is}\:\mathrm{not}\:\mathrm{guaranteed}\:\mathrm{when}\:\mathrm{only}\:\mathrm{finite} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{terms}\:\mathrm{are}\:\mathrm{known}. \\ $$$$\mathrm{Also}\:\mathrm{see}\:\mathrm{quesion}\:\mathrm{3003},\:\mathrm{where}\:\mathrm{two}\:\mathrm{different} \\ $$$$\mathrm{formulas}\:\mathrm{can}\:\mathrm{generate}\:\mathrm{all}\:\mathrm{known}\:\mathrm{terms} \\ $$$$\mathrm{correctly}\:\mathrm{but}\:\mathrm{differ}\:\mathrm{on}\:\mathrm{next}\:\mathrm{term}. \\ $$

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