# Arithmetic and Geometric Progressions

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## AP: Mean

a, b, c, d and e are 5 distinct numbers that from an arithmetic progression. They are not necessarily consecutive terms but form the first 5 terms of the AP. It is known that c is the arithmetic mean of a and b, and d is the arithmetic mean of b anc c. Which of the following statements are true?

i. Average of all 5 terms put together is c.
ii. Average of d and e is not greater than average of a and b.
iii. Average of b and c is greater than average of a and d.

1. i and ii only
2. ii and iii only
3. all three statements are true
4. i and iii only

Choice A. i and ii only

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## Detailed Solution

I. c is the arithmetic mean of a and b => c lies in between a and b. And it lies exactly in between the two terms. As in the number of terms between a and c should be equal to number of terms between b and c.

a, c, b could be the
1st 2nd and 3rd terms respectively or
1st, 3rd and 5th respectively, or
2nd, 3rd, 4th respectively, or
3rd, 4th, 5th respectively.

The terms could also be the other way around. As in, b, c, a could be the 1st 2nd and 3rd terms respectively, or the 1st, 3rd and 5th respectively, and so on. This is a very simple but very powerful idea.

II. Now, d is the arithmetic mean of b and c. => d lies between b and c. Using statements I and II we can say that a, c, b have to be 1st, 3rd and 5th or 5th, 3rd and 1st as there is an element between b and c also.

So, c is the third term. a and b are 1st and 5th in some order.

b ___ c ___ a or a ___ c ___ b

d is the arithmetic mean of b and c.

Possible arrangements are:

b     d   c   e   a    or a   e   c    d   b

Statement (I): The average of all 5 terms put together is c. c is the middle term. So this is true.

Statement (II): the average of d and e is not greater than average of a and b. Average of a, b is c. d and e are the 2nd and 4th terms of this sequence (in some order). So, their average should also be equal to c. So, both these are equal. So, this statement is also true.

Statement (III): The average of b and c is greater than average of a and d. The average of b and c is d. The average of a and d could be greater than or less than d. So, this need not be true.

Correct Answer: i and ii are true

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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.