Question

If $11$ A.M.'s are inserted between $28$ and $10$ , then find the number of integral A.M.'s.

Answer 1

Answer: 5

Let the $11$ A.M.'s be $A_1,A_2,\dots ,A_{11}$.
Then, $28,A_1,A_2,\dots ,A_{11},10$ are in A.P.
$d=\frac{10-28}{12}=-\frac{3}{2}$
$\Rightarrow A_i=28+id=28-\frac{3}{2}i,i=1,2,\dots ,11$
Clearly, $A_2,A_4,A_6,A_8,A_{10}$ are integral A.M.'s
$\Rightarrow$ Number of integral A.M.'s $=5$