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a/b , ab , a−b , a+b

Above shows real numbers that belong to an arithmetic progression in order. Find the next term of this sequence.

In the question I was able to come up with different answers like find d and using formula of of a + (n-1)*d Then another answer is 2(a+b) = a-b + variable

But I guess there needs to be a term which is simple Got any Ideas?

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    $\begingroup$ Third difference is $2b.$ What happens if this is equated to first and second differences, and try to solve? $\endgroup$
    – coffeemath
    Commented Jun 28, 2019 at 6:17

1 Answer 1

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Since they belong to an AP, then suppose the first term of the AP is given by $A$ and the common difference is $d$. Thus, we have the following equations:

$$ a/b = A \\ ab = A + d \\ a - b = A + 2d \\ a + b = A + 3d \\ $$

Solving the above system yields: $ a = -9/8, b = -3/5, A = 15/8, d = -6/5$.

Thus, the next term would be $A + 4d = \boxed{-\frac{117}{40}}$

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