Question

# Find a, b and c such that the following numbers are in A.P. : a, 7, b, 23, c.

Solution

## We have  $$d_1 = a_2 - a_1 = 7 - a$$ $$d_2 = a_3 - a_2 = b - 7$$ $$d_3 = a_4 - a_3 = 23 - b$$ $$d_4 = a_5 - a_4 = c - 23$$ As list of numbers is in A.P.,so $$d_1 = d_2 = d_3 = d_4$$ Now. $$d_2 = d_3$$$$\Rightarrow \ \ b - 7 = 23 - b$$ $$b + b = 30$$ $$\Rightarrow \ \ 2b = 30$$$$\Rightarrow \ \ b - 15$$ Now, $$d_2 = d_1$$$$\Rightarrow \ \ b - 7 = 7 - a$$$$\Rightarrow \ \ 15 - 7 = 7 - a$$$$\Rightarrow \ \ 8 = 7 - a$$$$a = 7 - 8 = -1$$Now, $$d_4 = d_2$$$$\Rightarrow \ \ c - 23 = b - 7$$$$\Rightarrow \ \ c = 23 + 15 - 7 = 38 - 7$$$$\Rightarrow \ \ c = 31$$Hence, $$a = -1, b = 15$$ and $$c = 31$$.Mathematics

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