A,A=1/56+1/72+1/90+1/110+1/132+1/156 B,B=1/1.6+1/6.11+1/11.16+...+1/26.31 Help

a)A=1/56+1/72+1/90+1/110+1/132+1/156

   A=1/7.8+1/8.9+1/9.10+1/10.11+1/11.12+1/12.13

   A=1/7-1/8+1/8-1/9+1/9-1/10+1/10-1/11+1/11-1/12+1/12-1/13

   A=1/7-1/13

  A=6/91

b)B=1/1.6+1/6.11+1/11.16+...+1/26.31

D=1/5(1−1/31)

D=6/31

Answer

\(\begin{array}{l}
a, A = \dfrac{1}{56} + \dfrac{1}{72} + \dfrac{1}{90} + \dfrac{1}{110} + \dfrac{1}{132} + \dfrac{1}{156}\\
A = \dfrac{1}{7 . 8} + \dfrac{1}{8 . 9} + \dfrac{1}{9 . 10} + \dfrac{1}{10 . 11} + \dfrac{1}{11 . 12} + \dfrac{1}{12.13}\\
A = \dfrac{1}{7} - \dfrac{1}{8} + \dfrac{1}{8} - \dfrac{1}{9} + \dfrac{1}{9} - \dfrac{1}{10} + \dfrac{1}{10} - \dfrac{1}{11} + \dfrac{1}{11} - \dfrac{1}{12} + \dfrac{1}{12} - \dfrac{1}{13}\\
A = \dfrac{1}{7} - \dfrac{1}{13}\\
A = \dfrac{6}{91}\\
----------\\
b, B = \dfrac{1}{1 . 6} + \dfrac{1}{6 . 11} + \dfrac{1}{11 . 16} + ... + \dfrac{1}{26 . 31}\\
B = \dfrac{5}{5} . \left(\dfrac{1}{1 . 6} + \dfrac{1}{6 . 11} + \dfrac{1}{11 . 16} + ... + \dfrac{1}{26 . 31}\right)\\
B = \dfrac{1}{5} . \left(\dfrac{5}{1 . 6} + \dfrac{5}{6 . 11} + \dfrac{5}{11 . 16} + ... + \dfrac{5}{26 . 31}\right)\\
B = \dfrac{1}{5} . \left(1 - \dfrac{1}{6} + \dfrac{1}{6} - \dfrac{1}{11} + \dfrac{1}{11} - \dfrac{1}{16} + ... + \dfrac{1}{26} - \dfrac{1}{31}\right)\\
B = \dfrac{1}{5} . \left(1 - \dfrac{1}{31}\right)\\
B = \dfrac{1}{5} .  \dfrac{30}{31}\\
B = \dfrac{6}{31}\\
\end{array}\)