If the symbol [x] denotes the greatest integer less than or equal to x, then the value of :
$$\left[ {\frac{1}{4}} \right]$$ $$ + $$ $$\left[ {\frac{1}{4} + \frac{1}{{50}}} \right]$$   $$ + $$ $$\left[ {\frac{1}{4} + \frac{2}{{50}}} \right]$$   $$ + $$ $$....$$ $$ + $$ $$\left[ {\frac{1}{4} + \frac{{49}}{{50}}} \right]$$

Clearly, each of the 38 terms
$$\left[ {\frac{1}{4}} \right], \left[ {\frac{1}{4} + \frac{1}{{50}}} \right], \left[ {\frac{1}{4} + \frac{2}{{50}}} \right], $$      $$ ..... $$ $$ , \left[ {\frac{1}{4} + \frac{{37}}{{50}}} \right]$$   has a value lying between 0 and 1,
While each one of the 12 terms
$$\left( {\frac{1}{4} + \frac{{38}}{{50}}} \right),$$   $$\left( {\frac{1}{4} + \frac{{39}}{{50}}} \right),$$   $$.....,$$ $$\left( {\frac{1}{4} + \frac{{49}}{{50}}} \right)$$   has a value lying between 1 and 2.

Hence, the given expression
= (0 × 38) + (1 × 12)
= 12