Binomial Theorem
Recursive Integral-Defined Sequence
nta_pyq_2023_apr
Grade 11
Question:
Let for $x\in\mathbb{R}$, $S_0(x)=x$, $S_k(x)=C_kx+k\int_0^x S_{k-1}(t)\,dt$ where $C_0=1$, $C_k=1-\int_0^1 S_{k-1}(x)\,dx$, $k=1,2,3,\ldots$. Then $S_2(3)+6C_3$ is equal to _______.
Step-by-Step Solution
Key Concept: Recursively compute: $C_1=\frac{1}{2}$, $S_1(x)=\frac{x}{2}+\frac{x^2}{2}$; $C_2=\frac{7}{12}$, $S_2(x)=\frac{7x}{12}+\frac{x^2}{2}+\frac{x^3}{3}$; $C_3=\frac{11}{24}$.
$S_2(3)+6C_3=\frac{61}{4}+\frac{11}{4}=18$.
Correct Answer: 18