S=43​+98​+1615​+...+50002499​

\(S = 1 - \frac{1}{4} + 1 - \frac{1}{9} + 1 - \frac{1}{16} + . . . + 1 - \frac{1}{5000}\)

\(S = \left(\right. 1 + 1 + 1 + . . . + 1 \left.\right) - \left(\right. \frac{1}{4} + + \frac{1}{9} + \frac{1}{16} + . . . + \frac{1}{5000} \left.\right)\)

\(S = 49 - \left(\right. \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + . . . + \frac{1}{5 0^{2}} \left.\right) < 49\)\(\left(\right. 1 \left.\right)\)

Lại có : 

\(\frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + . . . + \frac{1}{5 0^{2}} < \frac{1}{1.2} + \frac{1}{2.3} + \frac{1}{3.4} + . . . + \frac{1}{49.50}\)

\(= \frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + . . . + \frac{1}{49} - \frac{1}{50} = 1 - \frac{1}{50} < 1\)

\(\Rightarrow\)\(- \left(\right. \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + . . . + \frac{1}{5 0^{2}} \left.\right) > - 1\)

\(\Rightarrow\)\(S = 49 - \left(\right. \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + . . . + \frac{1}{5 0^{2}} \left.\right) > 49 - 1 = 48\)\(\left(\right. 2 \left.\right)\)

Từ (1) và (2) suy ra : 

\(48 < S < 49\)

Vậy S không là số tự nhiên 

Chúc các bạn học tốt nhé ! =))

S=43​+98​+...+25002499​

\(= \frac{2^{2} - 1}{2^{2}} + \frac{3^{2} - 1}{3^{2}} + . . . + \frac{5 0^{2} - 1}{5 0^{2}}\)

\(= \left(\right. 1 + 1 + . . . + 1 \left.\right) - \left(\right. \frac{1}{2^{2}} + \frac{1}{3^{2}} + . . . + \frac{1}{5 0^{2}} \left.\right)\)

\(= 49 - \left(\right. \frac{1}{2^{2}} + \frac{1}{3^{2}} + . . . + \frac{1}{5 0^{2}} \left.\right)\)

\(\frac{1}{2^{2}} < \frac{1}{1 \cdot 2} = 1 - \frac{1}{2}\)

\(\frac{1}{3^{2}} < \frac{1}{2 \cdot 3} = \frac{1}{2} - \frac{1}{3}\)

...

\(\frac{1}{5 0^{2}} < \frac{1}{49 \cdot 50} = \frac{1}{49} - \frac{1}{50}\)

Do đó: \(\frac{1}{2^{2}} + \frac{1}{3^{2}} + . . . + \frac{1}{5 0^{2}} < 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + . . . + \frac{1}{49} - \frac{1}{50} = 1 - \frac{1}{50}\)

=>\(\frac{1}{2^{2}} + \frac{1}{3^{2}} + . . . + \frac{1}{5 0^{2}} < 1\)

=>\(0 < \frac{1}{2^{2}} + \frac{1}{3^{2}} + . . . + \frac{1}{5 0^{2}} < 1\)

=>\(0 > - \left(\right. \frac{1}{2^{2}} + \frac{1}{3^{2}} + . . . + \frac{1}{5 0^{2}} \left.\right) > - 1\)

=>\(0 + 49 > - \left(\right. \frac{1}{2^{2}} + \frac{1}{3^{2}} + . . . + \frac{1}{5 0^{2}} \left.\right) + 49 > - 1 + 49\)

=>49>B>48

=>S không là số tự nhiên