$\text{a,b,c,d}$ and e are 5 numbers that forms an arithmetic progression. They are necessarily consecutive terms but form the first 5 terms of the A.P. it is known that c is the arithmetic mean if a and b and d is the arithmetic mean of b and 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 greater than average of a and b

c is the arithmetic mean of a and b $\Rightarrow$  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 or terms between b and c

$\therefore$  a, c, d could be 1^st, 2^nd, 3^rd terms respectively (or)

1^st ,3^rd and 5^th respectively (or) 2^nd, 3^rd, 4^th respectively (or) 3^rd, 4^th, 5^th respectively .

Now d is the arithmetic mean of b and c

$\Rightarrow$  d lies between b and c.

$\therefore$  a, c, d have to be 1^st, 3^rd and 5^th (or)5^th, 3^rd and 1^st terms respectively

$\therefore$  Possible arrangements are b, d, c, e, a (or) a, e, c, d, b

Averages of all 5 terms in both average is c

$\therefore$ So, 1^st statement is true.

Average of a, b is c and average of d and e also c

So, 2^nd statement is also true.

Average of b and c is d

So average of a and d could be greater than (or) lesser than d

$\therefore$  So, 3^rd statement need not be true.