Sequences and Series

[lovely_nancy]

could some one please guide me through this:

A sequence of numbers t? t? t? t? ... is formed by taking a starting value of t? and using the rule.
Tk+1 = tk"2( squared) – 2 for K = 1, 2, 3 ...
Determine whether the sequence converges, diverges or is periodic in the case when:
(a) t1 = 3
(b) t1 = 1

 

[tkhunny]

I would try running it out a few terms. Go ahead and get your hands dirty. It should be relateively obvious for each after 3 or 4 terms.

 

[lovely_nancy]

I would try running it out a few terms. Go ahead and get your hands dirty. It should be relateively obvious for each after 3 or 4 terms.

could you gimme an example please! Im just so stuck on how to go about it

 

[tkhunny]

t1 = 5
t2 = 5^2 - 2 = 25 - 2 = 23
t3 = 23^2 - 2 = 529 - 2 = 527
...

 

[mmm4444bot]

Tk+1 = tk"2( squared) – 2

Let's fix this notation.

The symbols T and t do not mean the same thing; use only one or the other.

The expressions k and k+1 are subscripts (counters).

t_(k + 1) = (t_k)^2 - 2

This formula shows that each element in the sequence is the square of the preceeding element less two.

Follow tkhunny's suggestion, and write out the first few elements in the sequence, one by one, using the Natural numbers for k.

Here's an example of what he means.

EG: The first element in the sequence is 5.

This means that t_1 = 5

Now, we let k take on the values 1, 2, and 3 (the first three Natural numbers).

When k = 1, the subscript k+1 is 2. This example gives t_k as 5 (remember that k is currently 1; hence, the notation t_k currently means t_1), so the formula above gives:

t_2 = (5)^2 - 2 = 23

When k = 2, the subscript k+1 becomes 3. We just found that t_k is now 23, so the formula gives:

t_3 = 23^2 - 2 = 527

When k = 3, the subscript k+1 = 4. We just found that t_k is now 527, so the formula gives:

t_4 = 527^2 - 2 = 277727

The sequence in my example begins with these four elements (t_1 through t_4):

5, 23, 527, 277727

Can you try tkhunny's suggestion now?