question_answer

Find the middle term of the A.P. 6, 13, 20, ??., 216.

Answer:

Given A.P. is 6, 13, 20, ???, 216

Here, \[~a=6,\text{ }d=13-6=20-13=7\]

Let n be the number of terms.

then                  \[{{T}_{n}}=a+(n-1)d\]

\[216=6+(n-1)7\]

\[216=6+7n-7\]

\[217=7n\]

\[\therefore n=31\]

and middle term is \[\frac{(n+1)th}{2}\] term i.e., 16th term

\[\therefore {{T}_{16}}=6+(16-1)7\]

\[=6+15\times 7\]

\[\therefore {{T}_{16}}=111\]

\[\therefore \] Middle term of the A.P. is 111.