# Arithmetic and Geometric Progressions

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## GP Median

Consider a, b, c in a G.P. such that |a + b + c| = 15. The median of these three terms is a, and b = 10. If a > c, what is the product of the first 4 terms of this G.P.?
1. 40000
2. 32000
3. 8000
4. 48000

Choice A. 40000

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## Detailed Solution

Median is the first term => Common ratio has to be negative. why?

Let us see why this is true.

When a > 0,
If r > 1, this will be an increasing G.P.
If r lies between (0, 1), this will be a decreasing G.P.

In both cases, the middle term will be the median. If a < 0, the order will be the other way around, but the middle term would still be the median.

If the middle term is not the median, we can say that r < 0. Now, let us go the solution

b = 10, a and c should be negative. Solution, a + b + c cannot be 15.

a + b + c = –15

${\frac{ 10 }{r} + 10 + 10r=-15 \Rightarrow \frac{2}{r} +2r=-5}$

Solving the quadratic, we will get r = ${\frac{ -1 }{2}}$ or -2.

The sequence is either – 5, 10, – 20 or – 20, 10, – 5.
a > c => the sequence has to be – 5, 10, – 20.
The product of the first 4 terms = – 5 × 10 × –20 × 40 = 40000.

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## More questions from Progressions

With some simple but very powerful ideas, one can cut down on a lot of working when it comes to progressions. For example, anchoring a progression around its middle term can be very useful. Reinforce these ideas with these questions.