1) \[1^{2}+3^{2}+5^{2}............+15^{2}= ?\]
Explanation: \[\frac{n(2n-1)(2n+1)}{3}\: n=\] एकूण संख्या
\[\frac{8 \times 15\times 17}{3}=680\]
2) \[1^{2}+3^{2}+5^{2}............+21^{2}= ?\]
Explanation: \[\frac{11 \times 21\times 23}{3}= 1771\]
3) \[\frac{1}{2^{1}}+\frac{1}{2^{2}}+\frac{1}{2^{3}}..........+\frac{1}{2^{9}}=?\]
Explanation: \[1-\frac{1}{2^{n}}\]
\[1-\frac{1}{2^{9}}=1-\frac{1}{512}=\frac{511}{512}\]
4) \[(1\times 2)+(2\times 3)+(3\times 4)+...........+(8\times 9)=9\]
Explanation: \[\frac{n(n+1)(n+2)}{3}\]
\[\frac{8\times 9\times 10}{3}=240\]
5) \[\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+........+\frac{1}{9\times 10}\]
Explanation:\[\frac{n}{n+1}\: \: \: \: \frac{9}{10}\]
\[\bigstar \, \, 2^{2}+4^{2}+......40^{2}=?\]
\[\bigstar \: \: 2^{2}(1^{2}+2^{2}.........20^{2})=4\times 2870=11480\]
6) \[2^{0}+2^{1}+ 2^{2}+2^{3}.........+21^{0}\]
Explanation:\[2^{n^{+1}}-1\: \: \: \: \: \: 2^{11}-1\]
= 2048 - 1 = 2047
7) \[\left ( 1-\frac{1}{2^{2}} \right )\left ( 1-\frac{1}{3^{2}} \right )\left ( 1-\frac{1}{4^{2}} \right )......\left ( 1-\frac{1}{10^{2}} \right )=?\]
Explanation: \[\left ( 1-\frac{1}{2} \right )\left ( 1+\frac{1}{2} \right )\left ( 1-\frac{1}{3} \right )\left ( 1+\frac{1}{3} \right )\left ( 1-\frac{1}{4} \right )\]
\[\left ( 1-\frac{1}{5} \right )........\left ( 1-\frac{1}{10} \right )\left ( 1+\frac{1}{10} \right )\]
\[\left \{ \left ( 1-\frac{1}{2} \right )\left ( 1-\frac{1}{3} \right )\left ( 1-\frac{1}{4} \right ) .....\left ( 1-\frac{1}{10} \right )\right \}\]
\[\left \{ \left ( 1+\frac{1}{2} \right )\left ( 1+\frac{1}{3} \right )........\left ( 1+\frac{1}{10} \right )\right \}\]
\[\left \{ \frac{1}{2}\times \frac{2}{3} \times \frac{3}{4}.......\frac{9}{10}\right \}\]
\[\left \{ \frac{3}{2}\times \frac{4}{3} \times \frac{5}{4}.......\frac{11}{10}\right \}\]
\[\frac{1}{10}\times \frac{11}{2}=\frac{11}{20}=0.55\]
8) \[\left ( \frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}.......\frac{1}{99\times 100} \right )\]
Explanation:\[\frac{n}{n+1}=\frac{99}{100}\]
9) \[\left ( 1-\frac{1}{3} \right )\left ( 1-\frac{1}{4} \right )\left ( 1-\frac{1}{5} \right )........\left ( 1-\frac{1}{10} \right )=?\]
Explanation: \[\frac{2}{3}\times \frac{3}{4}\times \frac{4}{5}\times.......\frac{9}{10}\times =\frac{2}{10}=0.2\]
Or
\[\frac{2}{n}=\frac{2}{10}=0.2\]10)\[\left ( 1-\frac{1}{3} \right )\left ( 1-\frac{1}{4} \right ).....\left ( 1-\frac{1}{100} \right )\]
Explanation: \[\frac{2}{100}=0.02\]
11) \[\left ( 2-\frac{1}{3} \right )\left ( 2-\frac{3}{5} \right )\left ( 2-\frac{5}{7} \right )....\left ( 2-\frac{97}{99} \right )=?\]
Explanation: \[\frac{5}{3}\times \frac{7}{5}\times \frac{9}{7}\times ........\frac{101}{99}\times =\frac{101}{3}\]