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Linear Recurrence Relations

Take your recursion skills to the next level. If you've got a recurrence relation but no computer, how can you find a closed form? What about asymptotic behavior? How fast do rabbits reproduce?

Calculating Initial Terms

         

Let \( \{x_n\} \) be the sequence \( 1, 5, 9, 13, \ldots. \)

If \( \{x_n\} \) can be defined as the recurrence relation \[ x_7 = a, \quad x_{n+1} = x_n + b, \] find the value of \( a + b. \)

The sequence \(X\) is defined by \(X_1 = 2\) and \(X_n = X_{n-1} + 3\) for \(n \geq 2.\) What is the value of \(X_{10}?\)

A sequence \(V\) satisfies \(V_1 = 5, V_2 = 2\) and \(V_n = V_{n-1} + V_{n-2}\) for \(n \geq 2.\) What is the value of \(V_6?\)

In the sequence \( a_{n},\) if \(a_{1} = 1, a_{2}= 5\) and \(2a_{k+1} = a_{k} + a_{k+2},\) what is \(a_{10}?\)

The sequence \(F\) is defined by \(F_1 = 1\) and \(F_n = 2F_{n-1}\) for \(n \geq 2.\) What is \(F_6?\)

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