TargetIIT

Q1 [im]http://codecogs.izyba.com/gif.latex?x_n%3D1%5E2-2%5E2+3%5E2-....+%28-1%29%5E%7Bn-1%7Dn%5E2[/im].
Find x[ss]101[/ss]

Q2 Find no of roots of
(2.4)[p]x[/p]=(2.6)[p]x[/p]-1
Graphically it appears that one soln...but i am loooking for algebraic soln...

Q3 If a,b,c ε R and (b-1)[p]2[/p]<ac then find no. of solns of eqns
[im]http://codecogs.izyba.com/gif.latex?%5Cbegin%7Bmatrix%7D%20ax_1%5E2+2bx_1+c%3D2x_2%5C%5C%20ax_2%5E2+2bx_2+c%3D2x_3%5C%5C%20ax_3%5E2+2bx_3+c%3D2x_4%5C%5C%20...................%5C%5C%20...................%5C%5C%20ax_n%5E2+2bx_n+c%3D2x_1%5C%5C%20%5Cend%7Bmatrix%7D[/im]

1)

Considering in pairs

[im]http://codecogs.izyba.com/gif.latex?%281-2%29%281+2%29+%283-4%29%283+4%29+%285-6%29%285+6%29.............%2899-100%29%2899+100%29+101%5E%7B2%7D[/im]

now taking -1 common in each bracket....we get

[im]http://codecogs.izyba.com/gif.latex?-1%281+2+3+4.........100%29+101%5E2[/im]

=[im]http://codecogs.izyba.com/gif.latex?-1%28%5Cfrac%7B100.101%7D%7B2%7D%29+101%5E2[/im]

=[im]http://codecogs.izyba.com/gif.latex?5151[/im]

1)[im]http://codecogs.izyba.com/gif.latex?x_%7B101%7D%20%3D%201%5E2-2%5E2+3%5E3-4%5E2+...+%2099%5E2-100%5E2+101%5E2%5C%5C%20%5C%5C%20x_%7B101%7D%3D%20%281+2%29%281-2%29+%283+4%29%283-4%29+...+%2899+100%29%2899-100%29%20+101%5E2[/im]

[im]http://codecogs.izyba.com/gif.latex?S%20%3D%20-%281+2+3+4+....+99+100%29%20+101%5E2%5C%5C%20%5C%5C%20%5C%20%5C%20%3D%20-%5Cfrac%7B%28100%29%28101%29%7D%7B2%7D%20+%20101%5E2%20%3D%205151[/im]

1 min ki der.....:P

Q2: Consider [im]http://codecogs.izyba.com/gif.latex?f%28x%29%20%3D%20%282.6%29%5Ex-%282.4%29%5Ex%20-1[/im]

We can see that f(x)<-1 for x<0

When x≥0:

Consider

[im]http://codecogs.izyba.com/gif.latex?f%27%28x%29%20%3D%20%282.6%29%5Ex%20%5Cln%20%282.6%29-%282.4%29%5Ex%20%5Cln%20%282.4%29%20%3E%20%5Cln%20%282.4%29%20%5Cleft%28%282.6%29%5Ex-%282.4%29%5Ex%20%5Cright%29%3E0[/im]

Hence there is at most one root.

Since f(x) is continuous,  x=2 is the only root

Q3: Is quite familiar to me somehow:

The typical equation looks like [im]http://codecogs.izyba.com/gif.latex?ax_i%5E2%20-%202bx_i%20+c%20%3D%202x_%7Bi+1%7D[/im]

(Here we consider x[ss]n+1[/ss] = x[ss]1[/ss])

This can be written as [im]http://codecogs.izyba.com/gif.latex?ax_i%5E2%20-%202%28b-1%29x_i%20+c%20%3D%202%28x_%7Bi+1%7D-x_i%29[/im]

WLOG a>0. Then each term on LHS>0

Hence [im]http://codecogs.izyba.com/gif.latex?%5Csum_%7Bi%3D1%7D%5En%20ax_i%5E2%20-%202%28b-1%29x_i%20+c%20%3E0[/im]

But [im]http://codecogs.izyba.com/gif.latex?%5Csum_%7Bi%3D1%7D%5En%202%28x_%7Bi+1%7D-x_i%29%20%3D%200[/im]

Thus no real solutions are possible

thx a lot prophet sir,eragon,avik [1]