1and sir, why had u taken an,an+1,an+2 as equal......i.e, equal to x ??
Question. previously u had told the way to find generative function.
But I am unable to find the generative function of ax-1 + ax+1 = √2ax.
I got stuck after this step,
f(x)= a1x+(a2-√2a1)x2(x-1/√2)2+1/2
How to find an after that???
urs answer was Acos(Ï€x/4).
Please explain the way out sir.
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7 Answers
341Instead of generating functions, I will give you an algorithmic approach known as annihilators which is derived from Gen Funcs and notions of continuity
We first adapt the equation into a sequence form and write it as a_{n+2}-\sqrt 2 a_{n+1}+a_n = 0
Derive the characteristic polynomial x^2-\sqrt 2 x+1 = 0
which has the roots e^{i \frac{\pi}{4}}, e^{-i \frac{\pi}{4}}
From this derive the sequence as A \left(e^{i \frac{\pi}{4}} \right)^n+ B \left(e^{-i \frac{\pi}{4}}\right)^n = Ae^{i n\frac{\pi}{4}} +Be^{-i \frac{n\pi}{4}}
Usually we will be given a0 and a1 to get at A and B
But here we dont really need them to arrive at the conclusion that f(x+8) = f(x)
If you let A = B, and extend using continuity etc. you get f(x) = A cos πx/4 as a function satisfying the given equation
1Sir, after finding the roots, what is the purpose of writing the equation containing A & B with the roots?
Moreover what is the relation of a0 ,a1 with A & B? How can we find A & B with the help of a0 & a1?
Why had u taken A=B????? It was not given.
Sir, these doubts still persists. Pls clarify.
1
62see what hari sir is saying is something like
\\S_n=F_0+F_1x+F_2x^2....+F_rx^r+..... \\S_nx=F_0x+F_1x^2+F_2x^3....+F_rx^{r+1}+..... \\S_n/x=F_0/x+F_1+F_2x....+F_rx^{r-1}+..... \\\text{Adding the above three, after multiplying by } -\sqrt{2}, 1, 1 \text{respectively}
Find the value of Sn
Now Fn is the coefficient of xn in your sum...
I hope i have answered some of your doubts?
1no nishant bhaiya, I had seen that method, but this question is not solvabe by that method........In this question, I am unable to find the coefficient of xn only........That shud come A cos(Ï€x4)............hari sir, can u clarify my above doubts about ur metod?
62\\S_n\left(1+x+1/x\right)=-\sqrt{2}F_0+F_0x+F_0/x+F_1 \\S_n\left(1+x+x^2\right)=(F_1-\sqrt{2})F_0x+F_0x^2+F_0
find the coefficient of x^n in this expansion