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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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