Sequences & Series
Sequences and Series
nta_pyq_2025_jan
Grade 11
Question:
Let $\langle a_{n}\rangle$ be a sequence such that $a_{0}=0$, $a_{1}=\dfrac{1}{2}$ and $2a_{n+2}=5a_{n+1}-3a_{n},\ n=0,1,2,\dots$ Then $\displaystyle\sum_{k=1}^{100}a_{k}$ is equal to:
$3a_{99}-100$
$3a_{100}-100$
$3a_{99}+100$
$3a_{100}+100$
Step-by-Step Solution
Key Concept: Characteristic equation $2x^{2}-5x+3=0$ has roots $1$ and $3/2$, so $a_{n}=A+B(3/2)^{n}$. Initial conditions give $a_{n}=-1+(3/2)^{n}$.
Characteristic: $2x^{2}-5x+3=0\Rightarrow x=1,\,\tfrac{3}{2}.$ So $a_{n}=A+B(3/2)^{n}.$
$a_{0}=A+B=0$ and $a_{1}=A+\tfrac{3}{2}B=\tfrac{1}{2}\Rightarrow B=1,\,A=-1.$ Hence $a_{n}=-1+(3/2)^{n}.$
$$\sum_{k=1}^{100}a_{k}=-100+\sum_{k=1}^{100}\!\left(\dfrac{3}{2}\right)^{k}=-100+\dfrac{(3/2)\bigl((3/2)^{100}-1\bigr)}{(3/2)-1}=-100+3\bigl((3/2)^{100}-1\bigr).$$
Since $(3/2)^{100}=a_{100}+1$:
$$=-100+3(a_{100}+1)-3=3a_{100}-100.$$
Correct Answer: 2