Question

If ${{\text{a}}_1},{{\text{a}}_2},{{\text{a}}_3},.............,{{\text{a}}_n}$ are in arithmetic progression and ${{\text{a}}_1}{\text{ + }}{{\text{a}}_4} + {{\text{a}}_7} + ............. + {{\text{a}}_{16}} = 114$, then ${{\text{a}}_1}{\text{ + }}{{\text{a}}_6} + {{\text{a}}_{11}} + {{\text{a}}_{16}}$ is equal to:
$
  {\text{A}}{\text{. 38}} \\
  {\text{B}}{\text{. 98}} \\
  {\text{C}}{\text{. 76}} \\
  {\text{D}}{\text{. 64}} \\
$

Answer 1

Hint – Use the property of A.P. In case of arithmetic progressions, the difference between two adjacent terms always remains constant throughout the entire progression.

Complete step by step answer:
Given Data,
${{\text{a}}_1}{\text{ + }}{{\text{a}}_4} + {{\text{a}}_7} + {{\text{a}}_{10}} + {{\text{a}}_{13}} + {{\text{a}}_{16}} = 114{\text{ - - - Equation 1}}$
Looking at the above series, we can make out that
${{\text{a}}_1}{\text{ + }}{{\text{a}}_{16}}{\text{ = }}{{\text{a}}_7} + {{\text{a}}_{10}}{\text{ = }}{{\text{a}}_{13}} + {{\text{a}}_4}$
On substituting, equation 1 now becomes
$
  {\text{3}}\left( {{{\text{a}}_1}{\text{ + }}{{\text{a}}_{16}}} \right) = 114 \\
   \Rightarrow {{\text{a}}_1}{\text{ + }}{{\text{a}}_{16}} = 38.{\text{ - - - Equation 2}} \\
$
We are supposed to find out the value of, ${{\text{a}}_1}{\text{ + }}{{\text{a}}_6} + {{\text{a}}_{11}} + {{\text{a}}_{16}}$
Here, we can observe ${{\text{a}}_1}{\text{ + }}{{\text{a}}_{16}}{\text{ = }}{{\text{a}}_{11}} + {{\text{a}}_6}$
$
   \Rightarrow {{\text{a}}_1}{\text{ + }}{{\text{a}}_6} + {{\text{a}}_{11}} + {{\text{a}}_{16}} = 2\left( {{{\text{a}}_1}{\text{ + }}{{\text{a}}_{16}}} \right) \\
   \Rightarrow {{\text{a}}_1}{\text{ + }}{{\text{a}}_6} + {{\text{a}}_{11}} + {{\text{a}}_{16}} = 2\left( {38} \right){\text{ (from equation 2)}} \\
   \Rightarrow {{\text{a}}_1}{\text{ + }}{{\text{a}}_6} + {{\text{a}}_{11}} + {{\text{a}}_{16}} = 76. \\
    \\
$
Hence Option C is the correct answer.

Note – In order to solve problems like this, always look out for similarities or repetitions in the given series. Make use of it to simplify the arithmetic progression, find out the value of a term or a set of terms, the rest can be found out by direct substitution.