Let's analyze the first two terms of the given sequence.
1+1/2+ 1/2,
1+1/2+ 1/2+1/2
Notice that the fraction 12 is changed to the expression 1/(2+ 12). Next, analyze the second and the third terms of the sequence.
1+1/2+1/2+ 1/2,
1+1/2+1/2+ 1/2+1/2
Fortunately, we have the same behavior as before. Therefore in the next terms of the sequence we should change the fraction 12 to 1/(2+ 12). Let's write the fourth and fifth term of the sequence!
Fourth:
1+1/2+1/2+1/2+ 1/2+1/2
Fifth:
1+1/2+1/2+1/2+1/2+ 1/2+1/2
Simplified Terms
Next, using a calculator we can get the simplified terms of the given sequence.
Term
Value
First
1.4
Second
About 1.4167
Third
About 1.4138
Fourth
About 1.4143
Fifth
About 1.4142
Limiting value
Since sqrt(2)≈ 1.414213..., it appears that the sequence approaches the value of sqrt(2).
Extra
Alternative way of finding the limiting value
We will justify more precisely that indeed our sequence should converge to sqrt(2). Since every next term in the sequence is created by replacing 12 by the expression 1/(2+ 12), the limiting value (x) should be in the following form.
x=1+1/2+1/2+1/2+1/2+1/2+...
Next, let's transform the expression in the following way.
x=1+1/2+1/2+1/2+1/2+1/2+...
x=1+1/1+1+1/2+1/2+1/2+1/2+...
Notice that, in the part of the denominator we also get the expression x!
x=1+1/1+1+1/2+1/2+1/2+1/2+...
x=1+1/1+x
Now, we will solve the last equation to find the value of x.
