Sequences & Series
AP — Common Difference Maximizing a Product
nta_pyq_2024_jan
Grade 11
Question:
In an A.P., the sixth term $a_6=2$. If $a_1a_4a_5$ is the greatest, then the common difference of the A.P. is equal to
$\dfrac{3}{2}$
$\dfrac{8}{5}$
$\dfrac{2}{3}$
$\dfrac{5}{8}$
Step-by-Step Solution
Key Concept: $a_6=a+5d=2\Rightarrow a=2-5d$. $a_1a_4a_5=(2-5d)(2-2d)(2-d)$. Differentiate w.r.t. $d$ and set to zero.
$d=8/5$ gives maximum of $a_1a_4a_5$.
Correct Answer: 2