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Question
Insert six A.M.s between 15 and −13.
Solution
Let \[A_1 , A_2 , A_3 , A_4 , A_5 , A_6\] be the 6 A.M.s between 15 and \[-\] 13.
Then, 15, \[A_1 , A_2 , A_3 , A_4 , A_5 , A_6\] and \[-\] 13 are in A.P. whose common difference is as follows:
d = \[\frac{- 13 - 15}{6 + 1}\] = \[- 4\]
\[A_1 = 15 + d = 15 + \left( - 4 \right) = 11\]
\[ A_2 = 15 + 2d = 15 + \left( - 8 \right) = 7\]
\[ A_3 = 15 + 3d = 15 + \left( - 12 \right) = 3\]
\[ A_4 = 15 + 4d = 15 + \left( - 16 \right) = - 1\]
\[ A_5 = 15 + 5d = 15 + \left( - 20 \right) = - 5\]
\[ A_6 = 15 + 6d = 15 + \left( - 24 \right) = - 9\]
Hence, the required A.M.s are
\[11, 7, 3, - 1, - 5, - 9\].
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