Sequences & Series
Counting increasing GPs with natural number common ratio
MMTS_Full_Test_18
Grade 12
Question:
The number of three-term increasing geometrical progressions comprising distinct natural numbers less than or equal to 100, with common ratio as a natural number, is
(A) 106
(B) 53
(C) 47
(D) 104
Step-by-Step Solution
Key Concept: Write the GP as $a, ar, ar^2$ with $r\geq2$ (natural number, $r>1$ for increasing). Need $ar^2\leq100$, i.e., $a\leq100/r^2$. Count using floor function.
$\sum_{r=2}^{10}\lfloor100/r^2\rfloor = 25+11+6+4+2+2+1+1+1=53$.
Correct Answer: (B) 53