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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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