The terms 3^x - 1 , 3^(2x + 2), 5*3^x + 1 are the terms of an a.p. Determine the value of x .

We'll have to determine x, so that the given terms to be
the consecutive terms of an arithmetic series.

Now, we'll
apply the rule of the consecutive terms of an arithmetical progression. According to the
rule, the middle term is the arithmetical mean of the joined
terms.

3^(2x + 2) = ( 3^x - 1 + 5*3^x
+1)/2

Eliminating like terms, from the brackets, from the
right side, we'll get:

3^(2x + 2) = ( 3^x +
5*3^x)/2

Now, we'll factorize by 3^x, to the right
side:

3^(2x + 2) =
3^x*(5+1)/2

3^(2x + 2) =
3*3^x

3^(2x + 2) =
3^(x+1)

Because the bases are matching, we'll apply the one
to one property:

2x + 2 = x +
1

We'll isolate x to the left
side:

2x - x = 1 - 2

x =
-1

So, the consecutive terms of the A.P.
are:

1/3 - 1 , 3^0, 5/3 +
1

-2/3 , 1 ,
8/3