TargetIIT

A recurrence satisfies [im]http://codecogs.izyba.com/gif.latex?a_%7Bn+1%7D%20%3D%20a_n%20+%20%5Csqrt%7Ba_n+a_%7Bn+1%7D%7D[/im]

a[ss]1[/ss] =1

Obtain an explicit formula for a[ss]n[/ss] i.e. find a function f(n) such that a[ss]n[/ss] = f(n)

edited: first term provided now, sry.

any hint ?
[hide]sir , how to solve non linear recurrence , like this one ,  i mean for linear u told one techonique [/hide]

Can we wait a day for the hint?

[im]http://codecogs.izyba.com/gif.latex?a_n%3D%5Cfrac%7Bn%28n+1%29%7D%7B2%7D[/im]

xyz plz post the soln

i found first few terms of series
thjey came out to be
1,3,6,10
so
a[ss]n[/ss]-a[ss]n-1[/ss]=ln+c=linear
so
an=quadratic in n
a[ss]n[/ss]=ln[p]2[/p]+mn+c , now put
a1=1
a2=3
a3=6
we l=1,m=1 c=0
hence a[ss]n[/ss]=n(n+1)/2

not a strong proof though

[im]http://codecogs.izyba.com/gif.latex?f%281%29%3D1[/im]

[im]http://codecogs.izyba.com/gif.latex?f%282%29%3Df%281%20%29+2%3D3[/im]

[im]http://codecogs.izyba.com/gif.latex?f%283%29%3Df%282%20%29+3%3D6[/im]

Induction gives [im]http://codecogs.izyba.com/gif.latex?f%28n%29%3Df%28n-1%29+n%20%3Df%28n-2%29+%28n-1%29+n%3Df%28n-3%29+%28n-2%29+%28n-1%29+n[/im]

Continuing like this, we finally have [im]http://codecogs.izyba.com/gif.latex?f%28n%29%3Df%281%29+2+3+4+...+n%3D%5Cfrac%7Bn%28n+1%29%7D%7B2%7D[/im].

Happy xyz?

[im]http://codecogs.izyba.com/gif.latex?a_%7Bn+1%7D%3Da_%7Bn%7D+%5Csqrt%7Ba_%7Bn+1%7D+a_%7Bn%7D%7D[/im]
[im]http://codecogs.izyba.com/gif.latex?%5CRightarrow%20%28%5Csqrt%7Ba_%7Bn+1%7D+a_%7Bn%7D%7D%29%5E2-%5Csqrt%7Ba_%7Bn+1%7D+a_%7Bn%7D%7D-2a_%7Bn%7D%3D0[/im]
[im]http://codecogs.izyba.com/gif.latex?%5CRightarrow%20%5Csqrt%7Ba_%7Bn+1%7D+a_%7Bn%7D%7D%3D%5Cfrac%7B1%5Cpm%20%5Csqrt%7B1+8a_%7Bn%7D%7D%7D%7B2%7D[/im]
[im]http://codecogs.izyba.com/gif.latex?%5CRightarrow%20a_%7Bn+1%7D+a_%7Bn%7D%3D%5Cfrac%7B1+1+8a_%7Bn%7D%5Cpm%202%5Csqrt%7B1+8a_%7Bn%7D%7D%7D%7B4%7D[/im]
[im]http://codecogs.izyba.com/gif.latex?%5CRightarrow%20a_%7Bn+1%7D%3D%5Cfrac%7B2+4a_%7Bn%7D%5Cpm%202%5Csqrt%7B1+8a_%7Bn%7D%7D%7D%7B4%7D[/im]
[im]http://codecogs.izyba.com/gif.latex?%5CRightarrow%20a_%7Bn+1%7D%3Da_%7Bn%7D+%5Cfrac%7B2%5Cpm%202%5Csqrt%7B1+8a_%7Bn%7D%7D%7D%7B4%7D[/im]
[im]http://codecogs.izyba.com/gif.latex?%7B%5Ccolor%7Bred%7D%5CRightarrow%20a_%7Bn+1%7D%3Da_%7Bn%7D+%5Cfrac%7B1%5Cpm%20%5Csqrt%7B1+8a_%7Bn%7D%7D%7D%7B2%7D%7D[/im]

does that help much????[17][17][17]

TRY THIS QUESTION:
A recurrence satisfies
[im]http://codecogs.izyba.com/gif.latex?a_%7Bn+1%7D%3D%20a_%7Bn%7D+%5Csqrt%7B1+a_%7Bn%7D%7D[/im]

Obtain an explicit formula for [im]http://codecogs.izyba.com/gif.latex?a_%7Bn%7D[/im] i.e. find a function f(n) such that [im]http://codecogs.izyba.com/gif.latex?a_%7Bn%7D%3DF%28n%29[/im]
(given that [im]http://codecogs.izyba.com/gif.latex?a_%7B0%7D%3D0[/im])

didn't understand soumik.

hey soumik mind ur attitude[16]
after assuming a[ss]n[/ss]=n(n+1)/2
directly put it in recurrence relation ,u will get it true for a[ss]n+1[/ss]