In a series in AP if two terms are 18 and 32.How many minimum terms a, b, c… would be required between the two to insure that a, b, c… are unique?

In an AP, the mth and nth terms are given
by:

am = a1+(m-1)d, where a1 is the first term and d is the
common diffrence between the consecutive terms.

an =
a1+(n-1)d.

m is assumed to be less than
m.

The terms between am and an is
n-m.

Therefore , to have  n-m distinct terms between an and
am ,  we should have  the condition: (an-am)/d = n-m -1 , for a particular  common
difference d which should not be zero.

Therefore if  am =
18 and an = 32 and there are r different terms between 18 and
32.

d = (an-am)/(r+1) =
(32-18)/(r+1).

in particular if r = 2, then 2 terms are in
beween 18 and 32. We have only 2 terms between 18 and 32, then d=(32-18)/3 = 14/3 , so
that the between terms are 18+14/3 = 22 2/3. and 18+28/3 = 27
1/3.

If