TargetIIT

[b]1)[/b]find the nth term of the sequence[im]http://alt1.mathlinks.ro/latexrender/pictures/2/0/4/204ce731395b7a900761580e85c5e88c4a8582d1.gif[/im]such that [im]http://alt2.mathlinks.ro/latexrender/pictures/f/2/f/f2fbe27f391bb75dffcfd99416ed3fccc0eba7f0.gif[/im]


[b]2)[/b]Find the nth term of the sequence[im]http://alt1.mathlinks.ro/latexrender/pictures/2/0/4/204ce731395b7a900761580e85c5e88c4a8582d1.gif[/im] such that

[im]http://alt2.mathlinks.ro/latexrender/pictures/b/4/2/b42cbd20b5e0d2f475610449f39622cae3cfe3ac.gif[/im]


[b]3)[/b]Find the nth term of the sequence [im]http://alt1.mathlinks.ro/latexrender/pictures/2/0/4/204ce731395b7a900761580e85c5e88c4a8582d1.gif[/im] such that[im]http://alt1.mathlinks.ro/latexrender/pictures/8/2/0/8205f0bcedeb81ad5669b858e562edb636c6bddc.gif[/im]  then Calculate  [im]http://alt1.mathlinks.ro/latexrender/pictures/8/8/e/88e5dcfdbbb7c3e7f462aee361ba63e81fe855fa.gif[/im]


[b]4)[/b]Find the n th term of the sequence [im]http://alt1.mathlinks.ro/latexrender/pictures/2/0/4/204ce731395b7a900761580e85c5e88c4a8582d1.gif[/im] such that  [im]http://alt1.mathlinks.ro/latexrender/pictures/7/6/1/7618bd2ebcac42c428d8662f50028222718de46c.gif[/im]

[b]5)[/b]Find the n th term of the sequence [im]http://alt1.mathlinks.ro/latexrender/pictures/2/0/4/204ce731395b7a900761580e85c5e88c4a8582d1.gif[/im] such that  [im]http://alt1.mathlinks.ro/latexrender/pictures/8/a/3/8a373cd425f6cdda7d1873796c01f818f09d591c.gif[/im]

ya akari u r correct :)

try oder ones also ....i added 5th one

btw i had an approch for 1st one ....but much lengthier than generating polynomials

arre yaar look at the q...it says for n≥2[1][1][1]

[im]http://codecogs.izyba.com/gif.latex?a_%7Bn+1%7D%20%3D%20%5Cfrac%7Ba_n%7D%7B2%7D%20+%20%5Cfrac%7Bn%5E2-2n-1%7D%7Bn%5E2%28n+1%29%5E2%7D%20%3D%20%5Cfrac%7Ba_n%7D%7B2%7D%20+%20%5Cfrac%7B2%7D%7B%28n+1%29%5E2%7D%20-%20%5Cfrac%7B1%7D%7Bn%5E2%7D[/im]

[im]http://codecogs.izyba.com/gif.latex?%5CRightarrow%20a_%7Bn+1%7D%20-%20%5Cfrac%7B2%7D%7B%28n+1%29%5E2%7D%3D%20%5Cfrac%7B1%7D%7B2%7D%20%5Cleft%28a_n%20-%20%5Cfrac%7B2%7D%7Bn%5E2%7D%20%5Cright%29[/im]

So if you define a sequence {b[ss]n[/ss]} as [im]http://codecogs.izyba.com/gif.latex?b_n%20%3D%20a_n%20-%20%5Cfrac%7B2%7D%7Bn%5E2%7D[/im] then this sequence obeys the recurrence [im]http://codecogs.izyba.com/gif.latex?b_%7Bn+1%7D%20%3D%20%5Cfrac%7Bb_n%7D%7B2%7D[/im]

Q4. na[ss]n[/ss] = (n-1)S[ss]n[/ss]
(n-1)a[ss]n-1[/ss] = (n-2)S[ss]n-1[/ss]

Subtracting,
n(a[ss]n[/ss] - a[ss]n-1[/ss]) + a[ss]n-1[/ss] = (n-1)a[ss]n[/ss] + S[ss]n-1[/ss]
=> a[ss]n[/ss] - (n-1)a[ss]n-1[/ss] = (n-1)a[ss]n-1[/ss]/(n-2)

=> a[ss]n[/ss] = (n-1)[p]2[/p]a[ss]n-1[/ss]/(n-2) = (n-1)[p]2[/p](n-2)(n-3)(n-4)....2.a[ss]2[/ss]
also 2a[ss]2[/ss] = 1 + 1 + a[ss]2[/ss]
=> a[ss]2[/ss] = 2

=> a[ss]n[/ss] = 2(n-1)(n-1)!

quite long i guess

5) Rewriting the recurrence as [im]http://codecogs.izyba.com/gif.latex?n%5E3%20a_n%20%3D%20%28n-1%29%5E3%20a_%7Bn-1%7D%20+%20n%28n-1%29[/im]

Let [im]http://codecogs.izyba.com/gif.latex?b_n%20%3D%20n%5E3%20a_n[/im]

Then we have the recurrence [im]http://codecogs.izyba.com/gif.latex?b_n%20-b_%7Bn-1%7D%20%3D%20n%28n-1%29[/im]

This can be summed up telescopically to arrive at

[im]http://codecogs.izyba.com/gif.latex?b_n-b_1%20=%20%5Csum_%7Bk%3D2%7D%5En%20k%5E2-k%20%3D%20%5Csum_%7Bk%3D1%7D%5En%20k%5E2-k%20%3D%20%5Cfrac%7Bn%28n%5E2-1%29%7D%7B3%7D[/im]

b[ss]1[/ss]=0 and hence [im]http://codecogs.izyba.com/gif.latex?a_n%20%3D%20%5Cfrac%7B%28n%5E2-1%29%7D%7B3n%5E2%7D[/im]

thanx for replies prophet sir ,asish and akari

:)

btw q2 is left [im]http://www.mathlinks.ro/images/smiles/wink.gif[/im]

continuing from prophet sir's # 10 post

[im]http://codecogs.izyba.com/gif.latex?b_%7Bn%7D%3D2b_%7Bn+1%7D[/im]

[im]http://codecogs.izyba.com/gif.latex?b_%7Bn-1%7D%3D2b_%7Bn%7D[/im]

[im]http://codecogs.izyba.com/gif.latex?b_%7Bn-2%7D%3D2b_%7Bn-1%7D%3D4b_%7Bn%7D%3D2%5E%7B2%7Db_%7Bn%7D[/im]

[im]http://codecogs.izyba.com/gif.latex?%5CRightarrow%20b_%7Bn-k%7D%3D2%5E%7Bk%7Db_%7Bn%7D[/im]

[im]http://codecogs.izyba.com/gif.latex?%5CRightarrow%20b_%7Bn-%28n-1%29%7D%3D2%5E%7B%28n-1%29%7Db_%7Bn%7D[/im]

[im]http://codecogs.izyba.com/gif.latex?%5CRightarrow%20b_%7B1%7D%3D2%5E%7B%28n-1%29%7Db_%7Bn%7D[/im]

[im]http://codecogs.izyba.com/gif.latex?%5CRightarrow%20b_%7Bn%7D%3D%20%5Cfrac%7Bb_%7B1%7D%7D%7B2%5E%7B%28n-1%29%7D%20%7D[/im]

[im]http://codecogs.izyba.com/gif.latex?also%5C%3B%5C%3B%5C%3B%20b_%7Bn%7D%3D%20a_%7Bn%7D-%5Cfrac%7B2%7D%7Bn%5E%7B2%7D%7D[/im]

[im]http://codecogs.izyba.com/gif.latex?b_%7B1%7D%3D%20a_%7B1%7D-%5Cfrac%7B2%7D%7B1%7D%3D%20-1[/im]

[im]http://codecogs.izyba.com/gif.latex?a_%7Bn%7D-%5Cfrac%7B2%7D%7Bn%5E%7B2%7D%7D%3D%20%5Cfrac%7B-1%7D%7B2%5E%7Bn-1%7D%7D[/im]

[im]http://codecogs.izyba.com/gif.latex?a_%7Bn%7D%3D%5Cfrac%7B2%7D%7Bn%5E%7B2%7D%7D+%5Cfrac%7B-1%7D%7B2%5E%7Bn-1%7D%7D%3D%20%5Cfrac%7B2%5E%7Bn%7D-n%5E%7B2%7D%7D%7B2%5E%7B%28n-1%29%7Dn%5E%7B2%7D%7D[/im]

ya qwerty thats correct :)

btw u can do this

[im]http://latex.codecogs.com/gif.latex?\\\texttt{after%20\;%20u\;%20got\;%20this}\\%20a_{n+1}-\frac{2}{(n+1)^{2}}={1\over%202}\left(a_{n}-\frac{2}{n^{2}}\right)\\%20\Rightarrow%20\prod_{k=1}^{k=n-1}\left%20(%20a_{k+1}-\frac{2}{(k+1)^{2}}%20\right%20)=\prod_{k=1}^{k=n-1}\left%20(%20{1\over%202}\left(a_{k}-\frac{2}{k^{2}}\right)%20\right%20)\\\\\Rightarrow%20a_{n}-\frac{2}{n^2}=\frac{1}{2^{n-1}}\left%20(%20a_{1}-\frac{2}{{1}^{2}}%20\right%20)[/im]

lol ya [5] ,

for 1st one oder way

[im]http://codecogs.izyba.com/gif.latex?a_%7Bn+1%7D%3D2a_%7Bn%7D+1%5CLongleftrightarrow%20a_%7Bn+1%7D+1%3D2%28a_%7Bn%7D+1%29[/im]

[im]http://codecogs.izyba.com/gif.latex?%5Ctherefore%20a_%7Bn%7D+1%3D2%5E%7Bn-1%7D%28a_%7B1%7D+1%29%3D3%5Ccdot%202%5E%7Bn-1%7D%2C[/im]

threfore giving [im]http://codecogs.izyba.com/gif.latex?a_%7Bn%7D%3D3%5Ccdot%202%5E%7Bn-1%7D-1%5C%20%28n%5Cgeq%201%29[/im]