Sequences & Series
Telescoping Series
MMTS_Full_Test_05
Grade 12
Question:
Statement 1: $\sum_{k=1}^{\infty}\dfrac{6^k}{(3^k-2^k)(3^{k+1}-2^{k+1})}=2$. Statement 2: $\sum_{k=1}^{n}[k^3-(k-1)^3]=n^3$ for any natural number $n$.
Both S1 and S2 are true
Both S1 and S2 are false
S1 is true, S2 is false
S1 is false, S2 is true
Step-by-Step Solution
Key Concept: Telescoping for S1; telescoping sum for S2
S1: telescopes to 2. S2: telescopes to $n^3$. Both true.
Correct Answer: 1