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Q.
If $11$ A.M.'s are inserted between $28$ and $10$ , then find the number of integral A.M.'s.
Sequences and Series
Solution:
Let the $11$ A.M.'s be $A_{1}, A_{2}, \ldots, A_{11}$.
Then, $28, A_{1}, A_{2}, \ldots, A_{11}, 10$ are in A.P.
$d=\frac{10-28}{12}=-\frac{3}{2} $
$\Rightarrow A _{i}=28+i d=28-\frac{3}{2} i, i=1,2, \ldots, 11$
Clearly, $A _{2}, A _{4}, A _{6}, A _{8}, A _{10}$ are integral A.M.'s
$\Rightarrow$ Number of integral A.M.'s $=5$