Menu Close

1-4-5-16-17-20-what-is-the-68th-term-in-this-sequnce-




Question Number 28012 by JI Siam last updated on 18/Jan/18
1,4,5,16,17,20....... what is the 68th term in this sequnce?
$$\mathrm{1},\mathrm{4},\mathrm{5},\mathrm{16},\mathrm{17},\mathrm{20}…….\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{68th}\:\mathrm{term}\:\mathrm{in}\:\mathrm{this}\:\mathrm{sequnce}? \\ $$
Commented by JI Siam last updated on 18/Jan/18
please give answer fast
$$\mathrm{please}\:\mathrm{give}\:\mathrm{answer}\:\mathrm{fast} \\ $$
Answered by ajfour last updated on 18/Jan/18
T_(r+4) =2(−1)^r −(2/3)r(r+1)(r+2)+               r(r+1)+7r+14  T_(68) =T_(64+4)        =2−(2/3)(64×65×66)+64×65               +7×64+14       =2−43×64×65+448+14       =−43×64×65+464       =−178416 .
$${T}_{{r}+\mathrm{4}} =\mathrm{2}\left(−\mathrm{1}\right)^{{r}} −\frac{\mathrm{2}}{\mathrm{3}}{r}\left({r}+\mathrm{1}\right)\left({r}+\mathrm{2}\right)+ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{r}\left({r}+\mathrm{1}\right)+\mathrm{7}{r}+\mathrm{14} \\ $$$${T}_{\mathrm{68}} ={T}_{\mathrm{64}+\mathrm{4}} \\ $$$$\:\:\:\:\:=\mathrm{2}−\frac{\mathrm{2}}{\mathrm{3}}\left(\mathrm{64}×\mathrm{65}×\mathrm{66}\right)+\mathrm{64}×\mathrm{65} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{7}×\mathrm{64}+\mathrm{14} \\ $$$$\:\:\:\:\:=\mathrm{2}−\mathrm{43}×\mathrm{64}×\mathrm{65}+\mathrm{448}+\mathrm{14} \\ $$$$\:\:\:\:\:=−\mathrm{43}×\mathrm{64}×\mathrm{65}+\mathrm{464} \\ $$$$\:\:\:\:\:=−\mathrm{178416}\:. \\ $$
Commented by ajfour last updated on 19/Jan/18
1 ,4,     5, 16,      17, 20,        5,...T series    3,     1,  11,       1,  3,   −15,...S series     −2,  10, −10,  2,  −18,... R series             12, −20, 12,−20,... Q series               −32,    32,−32,..  P series  P_r =32(−1)^r   Q_(r+1) −Q_r =P_r   so  Σ_(r=1) ^r (Q_(r+1) −Q_r )=Σ_(r=1) ^r 32(−1)^r   ⇒ Q_(r+1) −Q_1 =16[(−1)^r −1]  so  Q_(r+1) −12=16(−1)^r −16  Q_(r+1) =16(−1)^r −4  proceeding in a similar manner  Σ_(r=1) ^r (R_(r+2) −R_(r+1) )=Σ_(r=1) ^r Q_(r+1)   R_(r+2) =8(−1)^r −4r+2  Then  Σ_(r=1) ^r (S_(r+3) −S_(r+2) )=Σ_(r=1) ^r R_(r+2)   S_(r+3) =4(−1)^r −2r(r+1)+2r+7  and finally, after another such step  T_(r+4) =2(−1)^r −(2/3)r(r+1)(r+2)+              r(r+1)+7r+14 .
$$\mathrm{1}\:,\mathrm{4},\:\:\:\:\:\mathrm{5},\:\mathrm{16},\:\:\:\:\:\:\mathrm{17},\:\mathrm{20},\:\:\:\:\:\:\:\:\mathrm{5},…{T}\:{series} \\ $$$$\:\:\mathrm{3},\:\:\:\:\:\mathrm{1},\:\:\mathrm{11},\:\:\:\:\:\:\:\mathrm{1},\:\:\mathrm{3},\:\:\:−\mathrm{15},…{S}\:{series} \\ $$$$\:\:\:−\mathrm{2},\:\:\mathrm{10},\:−\mathrm{10},\:\:\mathrm{2},\:\:−\mathrm{18},…\:{R}\:{series} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{12},\:−\mathrm{20},\:\mathrm{12},−\mathrm{20},…\:{Q}\:{series} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:−\mathrm{32},\:\:\:\:\mathrm{32},−\mathrm{32},..\:\:{P}\:{series} \\ $$$${P}_{{r}} =\mathrm{32}\left(−\mathrm{1}\right)^{{r}} \\ $$$${Q}_{{r}+\mathrm{1}} −{Q}_{{r}} ={P}_{{r}} \\ $$$${so}\:\:\underset{{r}=\mathrm{1}} {\overset{{r}} {\sum}}\left({Q}_{{r}+\mathrm{1}} −{Q}_{{r}} \right)=\underset{{r}=\mathrm{1}} {\overset{{r}} {\sum}}\mathrm{32}\left(−\mathrm{1}\right)^{{r}} \\ $$$$\Rightarrow\:{Q}_{{r}+\mathrm{1}} −{Q}_{\mathrm{1}} =\mathrm{16}\left[\left(−\mathrm{1}\right)^{{r}} −\mathrm{1}\right] \\ $$$${so}\:\:{Q}_{{r}+\mathrm{1}} −\mathrm{12}=\mathrm{16}\left(−\mathrm{1}\right)^{{r}} −\mathrm{16} \\ $$$${Q}_{{r}+\mathrm{1}} =\mathrm{16}\left(−\mathrm{1}\right)^{{r}} −\mathrm{4} \\ $$$${proceeding}\:{in}\:{a}\:{similar}\:{manner} \\ $$$$\underset{{r}=\mathrm{1}} {\overset{{r}} {\sum}}\left({R}_{{r}+\mathrm{2}} −{R}_{{r}+\mathrm{1}} \right)=\underset{{r}=\mathrm{1}} {\overset{{r}} {\sum}}{Q}_{{r}+\mathrm{1}} \\ $$$${R}_{{r}+\mathrm{2}} =\mathrm{8}\left(−\mathrm{1}\right)^{{r}} −\mathrm{4}{r}+\mathrm{2} \\ $$$${Then}\:\:\underset{{r}=\mathrm{1}} {\overset{{r}} {\sum}}\left({S}_{{r}+\mathrm{3}} −{S}_{{r}+\mathrm{2}} \right)=\underset{{r}=\mathrm{1}} {\overset{{r}} {\sum}}{R}_{{r}+\mathrm{2}} \\ $$$${S}_{{r}+\mathrm{3}} =\mathrm{4}\left(−\mathrm{1}\right)^{{r}} −\mathrm{2}{r}\left({r}+\mathrm{1}\right)+\mathrm{2}{r}+\mathrm{7} \\ $$$${and}\:{finally},\:{after}\:{another}\:{such}\:{step} \\ $$$${T}_{{r}+\mathrm{4}} =\mathrm{2}\left(−\mathrm{1}\right)^{{r}} −\frac{\mathrm{2}}{\mathrm{3}}{r}\left({r}+\mathrm{1}\right)\left({r}+\mathrm{2}\right)+ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{r}\left({r}+\mathrm{1}\right)+\mathrm{7}{r}+\mathrm{14}\:. \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *