Question

Jared creates a number sequence that has a first term of 2 and a second of term of 5. Each term after the second is created by subtracting the term before the previous term from twice the previous term. He uses S (t) to denote term number t in his sequence. For example, S (1) =2 and S (2) = 5

Answer 1

The rule for the sequence is S(n) = 2S(n-1) - S(n-2)

Alternative form: S(n) = S(n-1) + 3

Step-by-step explanation:

In this problem, we know the first two terms of the sequence:

S(1) = 2

S(2) = 5

We are told that each term after the second is created by subtracting the term before the previous term from twice the previous term. In other words, if we call:

S(n) the current term

S(n-1) the previous term

S(n-2) the term before the previous term

This statement translates into the following sequence:

S(n) = 2S(n-1) - S(n-2)

Because we are subtracting the term before the previous term, S(n-2), from twice the previous term, 2S(n-1).

We can apply now the rule to find the first few terms of the sequence after S(1) and S(2):

$S(3) = 2S(2) - S(1) = 2(5)-2=10-2 = 8$

$S(4) = 2S(3) - S(2) = 2(8)-5=16-5 = 11$

$S(5) = 2S(4) - S(3) = 2(11)-8=22-8 = 14$

We notice also that each term of the sequence is just equal to the previous term plus 3, so the sequence can also be written as

S(n) = S(n-1) + 3

Learn more about sequences:

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