Insert 7 A.M.s between 2 and 17.

Question:

Insert 7 A.M.s between 2 and 17.

Solution:

Let $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}, A_{7}$ be the seven A.M.s between 2 and 17 .

Then, $2, A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}, A_{7}$ and 17 are in A.P. whose common difference is as follows:

$d=\frac{17-2}{7+1}$

$=\frac{15}{8}$

$A_{1}=2+d=2+\frac{15}{8}=\frac{31}{8}$

$A_{2}=2+2 d=2+\frac{15}{4}=\frac{23}{4}$

$A_{3}=2+3 d=2+\frac{45}{8}=\frac{61}{8}$

$A_{4}=2+4 d=2+\frac{15}{2}=\frac{19}{2}$

$A_{5}=2+5 d=2+\frac{75}{8}=\frac{91}{8}$

$A_{6}=2+6 d=2+\frac{45}{4}=\frac{53}{4}$

$A_{7}=2+7 d=2+\frac{105}{8}=\frac{121}{8}$

Hence, the required A.M.s are $\frac{31}{8}, \frac{23}{4}, \frac{61}{8}, \frac{19}{2}, \frac{91}{8}, \frac{53}{4}, \frac{121}{8}$.

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