Question

If for the series \(a_1, a_2, a_3, \ldots\), etc., \(a_{n+1} - a_n\) bears a constant ratio with \(a_n + a_{n+1}\), then \(a_1, a_2, a_3, \ldots\) are in:

• A. A.P.
• B. G.P.
• C. H.P.
• D. Any other series

Hint:

In problems where the difference between terms has a constant ratio with the sum of consecutive terms, the series is typically a harmonic progression (H.P.).

Answer 1

The Correct Option is C

Step 1: The difference between successive terms \( a_{n+1} - a_n \) has a constant ratio with the sum of successive terms \( a_n + a_{n+1} \). This is expressed as:

\[ \frac{a_{n+1} - a_n}{a_{n+1} + a_n} = k \]

where \( k \) is a constant.

Step 2: The equation can be rearranged:

\[ a_{n+1} - a_n = k(a_{n+1} + a_n) \]

Step 3: Simplify:

\[ a_{n+1} - a_n = k a_{n+1} + k a_n \] \[ a_{n+1} - k a_{n+1} = a_n + k a_n \] \[ a_{n+1}(1 - k) = a_n(1 + k) \]

Step 4: Solve for \( a_{n+1} \):

\[ a_{n+1} = \frac{1 + k}{1 - k} a_n \]

This is a characteristic of a series with a constant ratio between terms. This defines a harmonic progression (H.P.) because the general term \( a_n \) is inversely proportional to an arithmetic progression.