The third and fifth terms of an a.p are x+y respectively. find the 12th term.

To calculate the terms of an a.p., we'll use the formula
of a general term of an arithmetic sequence.

an = a1 +
(n-1)r, where:

a1 is the first
term

n is the number of
terms

r is the common
difference.

Now, we'll write a3, according to this
formula:

a3 = a1 + (3-1)r, where a3 = x (from
enunciation)

x = a1 + 2r
(1)

a5 = a1 + (5-1)r, where a5 = y (also, from
enunciation)

y = a1 + 4r
(2)

We'll subtract (2) from
(1):

a1 + 2r - a1 - 4r =
x-y

We'll eliminate like
terms:

-2r = x-y

We'll divide
by -2:

r =
(y-x)/2

So, the common difference is
(y-x)/2.

Now, we can find
a1.

a3 = a1 + 2r

x = a1 +
2(y-x)/2

x = a1 + y - x

a1 =
x-y+x

a1 = 2x -
y

Now, we have all the necessary elements to
calculate a12:

a12  = a1 +
11r

We'll substitute a1 and
r:

a12 = 2x-y + 11(y-x)/2

a12
= (4x - 2y + 11y - 11x)/2

We'll combine like
terms:

a12 = (-7x +
9y)/2