Question 21

Consider a function $$f(k)$$ defined for positive integers $$k = 1,2, ..$$ ; the function satisfies the condition $$f(1) + f(2) + .. = \frac{p}{p-1}$$. Where $$p$$ is fraction i.e. $$0 < p < 1$$. Then $$f(k)$$ is given by

Solution

$$\frac{-p}{1-p}$$ can be compared with sum of an infinite G.P. series i.e. $$\frac{a}{1-r}$$ (where a is first term and r is common ratio)
Hence here a=(-p)
and r = p
So k^th term will be = $$(-p) (p)^{(k-1)}$$

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CAT 1990 DILR Question

Consider a function $$f(k)$$ defined for positive integers $$k = 1,2, ..$$ ; the function satisfies the condition $$f(1) + f(2) + .. = \frac{p}{p-1}$$. Where $$p$$ is fraction i.e. $$0 < p < 1$$. Then $$f(k)$$ is given by

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