\(A=\dfrac{1}{15}+\dfrac{1}{21}+...+\dfrac{1}{105}\)
\(A=2\left(\dfrac{1}{30}+\dfrac{1}{42}+...+\dfrac{1}{210}\right)\)
\(A=2\left(\dfrac{1}{5.6}+\dfrac{1}{6.7}+...+\dfrac{1}{14.15}\right)\)
\(A=2\left(\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}+...+\dfrac{1}{14}-\dfrac{1}{15}\right)\)
\(A=2\left(\dfrac{1}{5}-\dfrac{1}{15}\right)=2.\dfrac{2}{15}=\dfrac{4}{15}\)
\(A=\dfrac{1}{15}+\dfrac{1}{21}+\dfrac{1}{28}+...........+\dfrac{1}{105}\)
\(=2\left(\dfrac{1}{30}+\dfrac{1}{42}+......+\dfrac{1}{210}\right)\)
\(=2\left(\dfrac{1}{5.6}+\dfrac{1}{6.7}+......+\dfrac{1}{14.15}\right)\)
\(=2\left(\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}+.....+\dfrac{1}{14}-\dfrac{1}{15}\right)\)
\(=2\left(\dfrac{1}{5}-\dfrac{1}{15}\right)\)
\(=2.\dfrac{2}{15}\)
\(=\dfrac{4}{15}\)