Sequence and Series
Harmonic Progression
AIEEE 2006
Grade 11
Question:
If $a_1, a_2, \dots , a_n$ are in H.P., then the expression $a_1a_2 + a_2a_3 + \dots + a_{n-1}a_n$ equals:
(1) $(n - 1)a_1a_n$
(2) $na_1a_n$
(3) $(n + 1)a_1a_n$
(4) $\frac{n(n - 1)}{2} a_1a_n$
Step-by-Step Solution
Key Concept: Since $1/a_k$ are in A.P. with common difference $d$, write $a_k - a_{k+1} = d \cdot a_ka_{k+1}$, so $a_ka_{k+1} = (a_k - a_{k+1})/d$. Sum telescopically.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)