Sequences: Find the Pattern
In a sequence, each term starting with the third onward is defined by the formula $$a_n = a_{n-1} - a_{n-2}$$, where $$n$$ is an integer greater than 2. If the first term of the sequence is 3 and the second term is 4, what is the value of the 70th term of the sequence?
Incorrect.
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Incorrect.
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Incorrect.
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Incorrect.
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You can use this fact to skip ahead 66 terms since 66 is divisible by 6. Therefore,
>$$a_{67}=a_1=3$$
>$$a_{68}=a_2=4$$
>$$a_{69}=a_3=1$$
>$$a_{70}=a_4=-3$$
Correct.
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Here are the first few terms of the sequence:
>$$a_1 = \color{blue}{3}$$
>$$a_2 = \color{blue}{4}$$
>$$a_3=4-3 = \color{blue}{1}$$
>$$a_4=1-4 = \color{blue}{-3}$$
>$$a_5=-3-1 = \color{blue}{-4}$$
>$$a_6=-4-(-3) = \color{blue}{-1}$$
>$$a_7=-1-(-4) = \color{blue}{3}$$
>$$a_8=3-(-1) = \color{blue}{4}$$
>$$a_9=4-3 = \color{blue}{1} =a_3$$ because they share the same 2 preceding terms.
Hence—
>$$a_{10}=a_4=-3$$, and so on.
The whole sequence is a formed by a repeating sequence of six numbers:
>$$3,~ 4,~ 1,~ {-3},~ {-4},~ {-1}$$
-4
-3
-1
3
4
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