SOLUTION: Help with special sequences? I'm stuck on these two questions and it would be greatly appreciated if you could help me! Even if its one question, I would really like to get the wh

Question 1005810: Help with special sequences?
I'm stuck on these two questions and it would be greatly appreciated if you could help me! Even if its one question, I would really like to get the whole concept to help prepare for my test! Thank you!!
1. Find the 4th term if the sequence in which a^1 = 2 and a^n+1 = -4a^n + 2.
2.Find the third iterate, x^3, of the function f(x) = 3x + 5 for an initial value of x^0 = 1.

Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
If , the common ratio of consecutive terms is
.
It is a geometric sequence, where each term is the one before multiplied by ,
so as for all geometric sequences, .
In particular,
.
Since and ,
.

The terms are iterations using

Because the problem asked for the term , with index ,
it was easy enough to calculate each of the terms from the previous term until I got to .
No need to get more complicated.
If the problem had asked for a term with a much higher index, such as the 10th term,
I would have found a "formula" to get that term directly, without calculating all the terms in between.

Since is the sum of a geometric sequence with first term and common ratio ,
that sum is , so

That last "formula" applied to , and with , yields
.
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