**Given:**

The sequence is
{2,22,222,...}
.

Here,
a1=2,a2=22 and a3=222
.

**Calculation:**

Obtain the formula for general term
an
of the sequence

The first term of the given sequence can be expressed as follows.

a1=212=21−12

The second term of the given sequence can be expressed as follows:

a2=2⋅212=(21+12)12 [By product rule : xm⋅xn=xm+n]=(232)12

Use Power rule
(xm)n=xm⋅n
and simplify the term as shown below.

a2=232⋅12=234=21−14=21−122

The third term of the given sequence can be expressed as follows.

a3=22⋅212=2(21+12)12 [By product rule : xm⋅xn=xm+n]=(2⋅(232)12)12=(2⋅234)12 [By power rule:(xm)n=xm⋅n ]
.

Again use Product rule and Power rule and simplify the term of the expression.

a3=(21+34)12=(274)12=278=21−123

Thus, the formula for general term
an
of the sequence is
an=21−12n
.

Obtain the limit of the sequence (the value of the term
an
as *n* tends to infinity).

limn→∞an=limn→∞21−12n=21−12∞=21−0=2

Therefore, the limit of the sequence is 2.