TS EAMCET · Maths · Basic of Mathematics
If \(N\) denotes the set of all positive integers and if \(f: N \rightarrow N\) is defined by \(f(n)=\) the sum of positive divisors of \(n\) then, \(f\left(2^k \cdot 3\right)\), where \(k\) is a positive integers, is
- A \(2^{k+1}-1\)
- B \(2\left(2^{k+1}-1\right)\)
- C \(3\left(2^{k+1}-1\right)\)
- D \(4\left(2^{k+1}-1\right)\)
Answer & Solution
Correct Answer
(C) \(3\left(2^{k+1}-1\right)\)
Step-by-step Solution
Detailed explanation
Given that \(f(x)=\) the sum of positive divisors of \(n\). \(\begin{aligned} \therefore f\left(2^k \cdot 3\right) & =3\left(1+2+2^2+2^3+\ldots+2^k\right) \\ & =3 \frac{\left(2^{-k+1}-1\right)}{2-1}=3\left(2^{-k+1}-1\right) \end{aligned}\)
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