Let a_n be a sequence such that a_0=0, a_1=12, and 2a_n+2=5a_n+1-3a_n for n=0,1,2,.

Question

Let $\{a_n\}$ be a sequence such that $a_0=0$, $a_1=\dfrac12$, and $2a_{n+2}=5a_{n+1}-3a_n$ for $n=0,1,2,\ldots$. Then $\displaystyle\sum_{k=1}^{100}a_k$ is equal to:

Accepted Answer

The characteristic equation $2x^2-5x+3=0$ gives $x=1,\ \tfrac32$, so $a_n = A+B\left(\tfrac32\right)^n$.

From $a_0=0$: $A+B=0$. From $a_1=\tfrac12$: $A+\tfrac32B=\tfrac12$. Solving: $A=-1$, $B=1$, so $a_n = -1+\left(\dfrac32\right)^n$.

$$\sum_{k=1}^{100}a_k = -100+\sum_{k=1}^{100}\left(\frac32\right)^k = -100+3\left[\left(\frac32\right)^{100}-1\right] = 3a_{100}-100.$$

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