The fundamental period of the function fx-x-2x-3x- - -nx-nn-12x- n- is where - - - is the greatest integer function

F(x)=[x]+[2x]+[3x]+… + [nx] n(n+1)2x, ∵ [x]+{x}=x f(x)=x {x}+2x {2x}+3x {3x}+… + nx {nx} n(n+1)2x f(x)=x(1+2+3+ … +n) n(n+1)x2 ({x}+{2x}+{3x}+… +{nx}) f(x)=xn(n ...

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Standard XII

Chemistry

Heat Capacity

The fundament...

Question

The fundamental period of the function $f (x) = [x] + [2 x] + [3 x] + \dots + [n x] - \frac{n (n + 1)}{2} x, n \in N,$ is $($ where $[.]$ is the greatest integer function $)$

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Solution

$f (x) = [x] + [2 x] + [3 x] + \dots + [n x] - \frac{n (n + 1)}{2} x,$
$∵ [x] + {x} = x$
$f (x) = x - {x} + 2 x - {2 x} + 3 x - {3 x} + \dots + n x - {n x} - \frac{n (n + 1)}{2} x$
$f (x) = x (1 + 2 + 3 + \dots + n) - \frac{n (n + 1) x}{2} - ({x} + {2 x} + {3 x} + \dots + {n x})$
$f (x) = x \frac{n (n + 1)}{2} - \frac{n (n + 1)}{2} x - ({x} + {2 x} + {3 x} + \dots + {n x})$
$f (x) = - ({x} + {2 x} + {3 x} + \dots + {n x})$

We know that period of ${x}$ is $1$
Hence, period of $f (x)$ is
$L . C . M . (1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{n})$
$= \frac{L . C . M . (1, 1, \dots, n times)}{H . C . F . (1, 2, 3, \dots, n)} = 1$

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Heat Capacity

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The fundamental period of the function f(x)=[x]+[2x]+[3x]+⋯+[nx]−n(n+1)2x, n∈N, is ( where [ . ] is the greatest integer function )

The fundamental period of the function $f (x) = [x] + [2 x] + [3 x] + \dots + [n x] - \frac{n (n + 1)}{2} x, n \in N,$ is $($ where $[.]$ is the greatest integer function $)$