Question
Insert $6$ numbers between $3$ and $24$ such that the resulting sequence is an A.P.

Answer

Let $A_1, A_2, A_3, A_4, A_5 $ and $A_6 $ be six numbers between $3$ and $24$ such that $3, A_1, A_2, A_3, A_4, A5, A_6, 24$ are in A.P. Here, $a = 3, b = 24, n = 8.$
Thus, $24 = 3 + (8 - 1) d,$ so that $d = 3.$
Therefore, $A1 = a + d = 3 + 3 = 6;$
$A_2 = a + 2d = 3 + 2 \times 3 = 9;$
$A_3 = a + 3d = 3 + 3 \times 3 = 12;$
$A_4 = a + 4d = 3 + 4 \times 3 = 15;$
$A_5 = a + 5d = 3 + 5 \times 3 = 18;$
$A_6 = a + 6d = 3 + 6 \times 3 = 21.$
Therefore,six numbers between $3$ and $24$ are $6, 9, 12, 15, 18$ and $21.$

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