Sequences & Series
Arithmetico-Geometric Sum
nta_pyq_2023_apr
Grade 11
Question:
If $(20)^{19}+2(21)(20)^{18}+3(21)^2(20)^{17}+\cdots+20(21)^{19}=k(20)^{19}$, then $k$ is equal to _____.
Step-by-Step Solution
Key Concept: Let $k=\sum_{r=1}^{20}r(\frac{21}{20})^{r-1}$. Multiply by $\frac{21}{20}$, subtract to get an AGP. Use $S=\frac{1-(21/20)^{20}}{(1-21/20)^2}-\frac{20(21/20)^{20}}{1-21/20}$.
$k=400$.
Correct Answer: 400