The document discusses an arithmetic progression (AP) of 5 distinct numbers - a, b, c, d and e. It is given that c is the arithmetic mean of a and b, and d is the arithmetic mean of b and c. The document analyzes which of the following statements must be true: 1) the average of all 5 terms is c, 2) the average of d and e is not greater than the average of a and b, and 3) the average of b and c is greater than the average of a and d. Through logical reasoning about the properties of APs, the document determines that statements 1 and 2 must be true, but statement 3 need not be true.

1. Progressions

2. Progressions
a, b, c, d and e are 5 distinct numbers that are 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, d is the arithmetic mean of b and c. Which of the following
statements are true?
A. Average of all 5 terms put together is c.
B. Average of d and e is not greater than average of a and b.
C. Average of b and c is greater than average of a and d.
(a) a and b only (b) b and c only
(c) a, b and c only (d) a and c only

3. Progressions
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.

4. Progressions
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.

5. Progressions
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.

6. Progressions
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.
__b_ _ d _ _c_ _e_ _a_ or __a_ _ e _ _c_ _ d _ _b_

7. Progressions
Possible arrangements are
__b_ _ d _ _c_ _e_ _a_ or __a_ _ e _ _c_ _ d _ _b_
A. Average of all 5 terms put together is c.
B. Average of d and e is not greater than average of a and b.
C. Average of b and c is greater than average of a and d.
(a) a and b only (b) b and c only
(c) a, b and c (c) a and c only

8. Progressions
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.
Answer choice (a)

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