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Properties of Multiplication of Integers

7 If the sum ...

Question

7 If the sum of all positive rational numbers n such that n +84n+1941 is an integer is 189/k,then find k.

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Solution

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N

f : N ⟶ N

f (n)

=

n

f (2^{k} .3)

^{'} k^{'}

n

k

n \geq \frac{k (k + 1)}{2}

(x_{1}, x_{2}, . . ., x_{k}), x_{1} \geq 1, x_{2} \geq 2, x_{k} \geq k

x_{1} + x_{2} + . . . + x_{k} = n

S_{n}

2^{n} < k < 2^{n - 1}

9

S_{n}

S_{n}

2^{n} < k < 2^{n + 1},

n \geq 1

S_{n}

n

k

n

n \sum k = 1 k^{3} {(\frac{^{n} C_{k}}{^{n} C_{k - 1}})}^{2} = \frac{n {(n + 1)}^{2} (n + 2)}{12}

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