By plotting the relation between n and an, one notices that as n increases, an increases. However this increase is not proportional to the increase of n, an seems to be increasing towards 1.62.
Once n reaches 28 an ceases to increase, remaining stable at 1.618033988749890.
This suggests that as n becomes very large an – an+1 = 0
As such, we can conclude that the exact value for this infinite surd is 1.618033988749890.
Consider another infinite surd:
Find the formula for an+1 in terms of a
a1 =
a2 = a2 =
a3 = a3 =
an+1 =
an =
Calculate the decimal values of the first ten terms of the sequence
a1 = 1.847759065
a2 = 1.9615705608
a3 = 1.9903694533
a4 = 1.9975909124
a5 = 1.9993976374
a6 = 1.9998494037
a7 = 1.9999623506
a8 = 1.9999905876
a9 = 1.9999976469
a10 = 1.9999994117
Using technology, plot the relation between n and an. Describe what you notice.
By plotting the relation between n and an, one notices that as n increases, an increases. However this increase is not proportional to the increase of n, an seems to be increasing towards 2.
Once n reaches 17 an ceases to increase, remaining stable at 2.
This suggests that as n becomes very large an – an+1 = 0
As such, we can conclude that the exact value for this infinite surd is 2.
Consider the general infinite surd :
Find the formula for an+1 in terms of a
a1 =
a2 = a2 =
a3 = a3 =
x is a quadratic equation we can therefore find the discriminant
> 0
Since is greater than 0 this quadratic equation has two distinct solutions.
and
x1 < 0 as an infinite surd can only be positive we can reject this answer.
Therefore
Some values of k that make the expression an integer are: 0, 2, 6, 12, 20
For to be an integer, has to be an odd perfect square.
One can notice that the value of the odd perfect square is a series with
Therefore:
y+1 is always an integer
The general statement that represents all the values of k for which the expression is an integer is with n any integer.
The general statement is valid for all integers and therefore has no limitations.
You can see from the steps above the process I took to find the general statement.