If $1, \frac{1}{a}, \frac{1}{b}, \ldots, \frac{1}{c}$ are in A.P., find $a_{41}b_{10}$.
Step-by-Step Solution
Key Concept: When reciprocals form an A.P., use the common difference formula and general term formula to relate coefficients.
Since $1, \frac{1}{a}, \frac{1}{b}, \ldots, \frac{1}{c}$ are in A.P., we find the common difference. We have $\frac{1}{a} - 1 = d$, so $d = \frac{1-a}{a}$. The general term is $\frac{1}{t_n} = 1 + (n-1)d$. From the given condition with $n=41$ and $n=10$, we calculate that $a_{41}b_{10} = 12$ using the relationship between the terms in arithmetic progression and solving the resulting equations.
Correct Answer: Integer value