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Area between Two Curves

Let function ...

Question

Let function $f_{n}$ be the number of way in which a positive integer n can be written as an ordered sum of several positive integers. For example, for $n = 3$ , $f_{3} = 3, s i n c e, 3 = 3, 3 = 2 + 1$ and $3 = 1 + 1 + 1$ . Then $f_{5} =$

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Solution

The correct option is A 7
According to Question

$5 = 5 \dots 1$

$5 = 4 + 1 \dots 2$

$5 = 3 + 2 \dots 3$

$5 = 3 + 1 + 1 \dots 4$

$5 = 2 + 1 + 1 + 1 \dots 5$

$5 = 2 + 2 + 1 \dots 6$

$5 = 1 + 1 + 1 + 1 + 1 \dots 7$

So the answer is $7$

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Similar questions

f (n)

n

1^{'} s

2^{'} s

f (4) = 5

4 = 2 + 2 = 2 + 1 + 1 = 1 + 2 + 1 = 1 + 1 + 2 = 1 + 1 + 1 + 1

N

n \in N,

f_{n} = (n + 1)^{1 / 3} - n^{1 / 3}

A = {n \in N; f_{n + 1} < \frac{1}{3 (n + 1)^{2 / 3}} < f_{n}}

f (n)

n

1^{'} s

2^{'} s

f (4) = 5

4 = 2 + 2 = 2 + 1 + 1 = 1 + 2 + 1 = 1 + 1 + 2 = 1 + 1 + 1 + 1

The sum of all positive integers n for which $\frac{1^{3} + 2^{3} . . . . + (2 n)^{3}}{1^{2} + 2^{2} + . . . n^{2}}$ is also an integer is.

n,

a (n) = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + . . . + \frac{1}{n} \cdot

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