Solve {l}{1/2^2+1/3^2+1/4^2}{+1/2004^2}

$$\left. \begin{array} { l } { \frac { 1 } { 2 ^ { 2 } } + \frac { 1 } { 3 ^ { 2 } } + \frac { 1 } { 4 ^ { 2 } } } \\ { + \frac { 1 } { 2004 ^ { 2 } } } \end{array} \right.$$

Evaluate

$$\frac{850615}{2008008}\approx 0.42361136$$

Solution Steps

Calculate 2 to the power of 2 and get 4.

$$\frac{1}{4}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\frac{1}{2004^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{1}{4}+\frac{1}{9}+\frac{1}{4^{2}}+\frac{1}{2004^{2}}$$

Least common multiple of 4 and 9 is 36. Convert \frac{1}{4} and \frac{1}{9} to fractions with denominator 36.

$$\frac{9}{36}+\frac{4}{36}+\frac{1}{4^{2}}+\frac{1}{2004^{2}}$$

Since \frac{9}{36} and \frac{4}{36} have the same denominator, add them by adding their numerators.

$$\frac{9+4}{36}+\frac{1}{4^{2}}+\frac{1}{2004^{2}}$$

Add 9 and 4 to get 13.

$$\frac{13}{36}+\frac{1}{4^{2}}+\frac{1}{2004^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{13}{36}+\frac{1}{16}+\frac{1}{2004^{2}}$$

Least common multiple of 36 and 16 is 144. Convert \frac{13}{36} and \frac{1}{16} to fractions with denominator 144.

$$\frac{52}{144}+\frac{9}{144}+\frac{1}{2004^{2}}$$

Since \frac{52}{144} and \frac{9}{144} have the same denominator, add them by adding their numerators.

$$\frac{52+9}{144}+\frac{1}{2004^{2}}$$

Add 52 and 9 to get 61.

$$\frac{61}{144}+\frac{1}{2004^{2}}$$

Calculate 2004 to the power of 2 and get 4016016.

$$\frac{61}{144}+\frac{1}{4016016}$$

Least common multiple of 144 and 4016016 is 4016016. Convert \frac{61}{144} and \frac{1}{4016016} to fractions with denominator 4016016.

$$\frac{1701229}{4016016}+\frac{1}{4016016}$$

Since \frac{1701229}{4016016} and \frac{1}{4016016} have the same denominator, add them by adding their numerators.

$$\frac{1701229+1}{4016016}$$

Add 1701229 and 1 to get 1701230.

$$\frac{1701230}{4016016}$$

Reduce the fraction \frac{1701230}{4016016} to lowest terms by extracting and canceling out 2.

$$\frac{850615}{2008008}$$

Factor

$$\frac{5 \cdot 170123}{2 ^ {3} \cdot 3 ^ {2} \cdot 167 ^ {2}} = 0.4236113601141031$$