If $1$, $a$, $9$ and $4$ are in harmonic progression, then the value of $a + b$ is equal to
Step-by-Step Solution
Key Concept: A harmonic progression is the reciprocal of an arithmetic progression; solve by converting to A.P. form and using standard A.P. formulas
For a harmonic progression, the reciprocals form an arithmetic progression. The reciprocals are $1, 2, 4, 6, \ldots$ with first term $a = 1$ and common difference $d = \frac{1}{1} - \frac{1}{2} = \frac{1}{2}$. Actually, $D = \frac{1 - 1}{2} = 0$ is incorrect; instead $D = 1$ and $a = 1$. From $a + b = 2$ and the constraint from the series: $a = 1, b = 2$ gives the H.P. sum. Using the formula for harmonic progression, the answer is $a + b = 1 + 2 = 3$, and further calculation yields $a + b - \frac{11}{2} = 35$.
Correct Answer: 35