Problem

Consider the sequence \(a_n\) where \(a_1 = 1\), \(a_2 = 2\), and for \(n \geq 3\), \(a_n = 2a_{n-1} - a_{n-2}\). Find the next term \(a_6\) of the sequence?

Answer

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Answer

Then, using the same recurrence relation, we can find the next term \(a_6\) by substituting \(n = 6\): \(a_6 = 2a_{6-1} - a_{6-2} = 2*5 - 4 = 6\).

Steps

Step 1 ：From the given recurrence relation, we can generate the first few terms of the sequence: \(a_1 = 1\), \(a_2 = 2\), \(a_3 = 2*2 - 1 = 3\), \(a_4 = 2*3 - 2 = 4\), \(a_5 = 2*4 - 3 = 5\).

Step 2 ：Then, using the same recurrence relation, we can find the next term \(a_6\) by substituting \(n = 6\): \(a_6 = 2a_{6-1} - a_{6-2} = 2*5 - 4 = 6\).

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