Home » Ask & Discuss » Mathematics
« Back to Discussion

Differential Calculus

Blazing goIITian

Joined:	23 Jun 2009
Post:	448

8 Dec 2009 14:10:29 IST

0 People liked this

413

If f(0) = 1 and f (1) = 2 and f(x) = 1/2{ f(x+1)+f(x+2) } , Then find f(100) =

Engineering Entrance , JEE Main , JEE Advanced , Mathematics , Differential Calculus

If f(0) = 1 and f (1) = 2 and f(x) = 1/2{ f(x+1)+f(x+2) } , Then find f(100) =

Comments (7)

The HellBlaizer

Hot goIITian

Joined: 31 Jan 2009

Posts: 156

8 Dec 2009 14:14:17 IST

Like 0 people liked this

http://www.goiit.com/posts/list/differential-calculus-hard-question-951552.htm

This question has already been discussed on the forum.Check the link above!!!!!!!!!!

shiva kumar

Cool goIITian

Joined: 5 Dec 2009

Posts: 54

8 Dec 2009 14:21:56 IST

Like 1 people liked this

I use a short-cut method called annihilators, but here's a good way to understand how to deal with such recursive definitions of sequencesWe have to find a general term for a sequence \{ a_n \} with the property 2a_n = a_{n+1} + a_{n+2} Consider the polynomial f(x) = a_1x + a_2x^2+...+a_k x^k+... i.e. a polynomial that has the terms of the sequence as its coefficients. This is called the generating polynomial or more generally the generating function of the sequence. This will enable us to obtain an explicit function for the sequence. We have: f(x) = a_1x + a_2x^2+a_3x^3...+a_k x^k+... \\ \\x f(x) = a_1 x^2 + a_2x^3+...+a_{k-1} x^k +... \\ \\-2x^2 f(x) = -2a_1x^3-...-2a_{k-2}x^k-..... Adding up the three expressions, we get f(x) (1+x-2x^2) = a_1x +(a_1+ a_2)x^2 or f(x)= \frac{a_1x}{(1+2x)(1-x)} +\frac{(a_1+ a_2)x^2}{(1+2x)(1-x)} \\ \\= \frac{1}{3}[a_1 x +(a_1+a_2)x^2]\left[\frac{2}{1+2x} + \frac{1} {1-x} \right] We can expand 1/(1+2x) and 1/(1-x) and the compare coefficients on both sides to obtain a_n = \frac{4}{3} - \frac{(-2)^n}{3} Substituting n=100 gives the value of a100

jeevan kumre

New kid on the Block

Joined: 7 Dec 2009

Posts: 2

8 Dec 2009 14:22:22 IST

Like 0 people liked this

f(100)=20100

shiva kumar

Cool goIITian

Joined: 5 Dec 2009

Posts: 54

8 Dec 2009 14:26:20 IST

Like 1 people liked this

4-2^100/3 this que is been done already

eragon24

Blazing goIITian

Joined: 14 May 2009

Posts: 697

8 Dec 2009 14:36:40 IST

Like 0 people liked this

checkthisout

and thisalso

FORESTS

Blazing goIITian

Joined: 23 Jun 2009

Posts: 448

8 Dec 2009 17:37:36 IST

Like 1 people liked this

please solve by short-cut method called annihilators and give me full explaination

FORESTS

Blazing goIITian

Joined: 23 Jun 2009

Posts: 448

8 Dec 2009 19:13:52 IST

Like 1 people liked this

THANKS DEEPAK

Quick Reply

Reply

	Some HTML allowed.
	Keep your comments above the belt or risk having them deleted.
	Signup for a avatar to have your pictures show up by your comment
	If Members see a thread that violates the Posting Rules, bring it to the attention of the Moderator Team