isi-entrance 2005 Q3

isi-entrance · India · solved Sequences and series, recurrence and convergence Summation of sequence terms

Let $f(a, b)$ be a function satisfying $f(a, b) = f(a, c) + f(c, b) - 2f(a,c)f(c,b)$ with $f(99, 100) = 1/3$. Find $f(1, 100)$.

$$f(1,100) = f(1,99) + f(99,100) - 2f(1,99)f(99,100)$$ $$= f(1,99) + 1/3 - 2/3 f(1,99) = 1/3 + 1/3 f(1,99)$$ Expanding recursively: $$= 1/3 + 1/3^2 + 1/3^3 + \cdots + 1/3^{99} \to 1/2$$

Let $f(a, b)$ be a function satisfying $f(a, b) = f(a, c) + f(c, b) - 2f(a,c)f(c,b)$ with $f(99, 100) = 1/3$. Find $f(1, 100)$.

Paper Questions

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