Question

Let $a_1, a_2, a_3, \dots$ be a sequence of real numbers. Let $s_n = a_1 + a_2 + \dots + a_n$.
If $2s_n = n(c + a_n)$ for some real number $c$ and for all $n = 1, 2, 3, \dots$, then which one of the following statements is Correct?

• A. $a_1, a_2, a_3, \dots$ is an Arithmetic Progression.
• B. $a_1, 2a_2, 3a_3, \dots$ is an Arithmetic Progression.
• C. $a_1, a_2, a_3, \dots$ is a Geometric Progression.
• D. $a_1, 2a_2, 3a_3, \dots$ is a Geometric Progression.

Hint:

The sum formula of an Arithmetic Progression is \(s_n = \frac{n}{2}(a_1 + a_n)\).
Recognizing this structure immediately when \(c = a_1\) saves significant algebraic effort.

Answer 1

The Correct Option is A

To solve this problem, we need to verify the condition provided and see which statement about the sequence is correct. We are given a sequence of real numbers \(a_1, a_2, a_3, \dots\) with the condition:

\(2s_n = n(c + a_n)\)

where \(s_n = a_1 + a_2 + \dots + a_n\) and this holds for all positive integers \(n\).

1. \(2s_n = n(c + a_n)\) implies \(s_n = \frac{n(c + a_n)}{2}\).
2. Substituting for \(s_n\):
   • For \(n = 1\): \(s_1 = a_1 = \frac{1}{2}(c + a_1)\) which simplifies to \(a_1 = \frac{1}{2}c + \frac{1}{2}a_1\). Solving gives \(a_1 = c\).
   • For \(n = 2\): \(s_2 = a_1 + a_2 = \frac{2}{2}(c + a_2)\) which simplifies to \(s_2 = c + a_2\).
   • Generalizing for \(n\): \(s_n = a_1 + a_2 + \dots + a_n = \frac{n}{2}(c + a_n)\).
3. Observe \(s_{n-1}\):
   • \(s_{n-1} = a_1 + a_2 + \dots + a_{n-1}\),
   • then \(s_n = s_{n-1} + a_n\).
4. Using the formula:
   \(a_1 + a_2 + \dots + a_{n-1} + a_n = \frac{n}{2}(c + a_n)\)
   \(a_1 + a_2 + \dots + a_{n-1} = \frac{n-1}{2}(c + a_{n-1})\).
5. Subtract the two equations:
   • \(a_n = \frac{n}{2}(c + a_n) - \frac{n-1}{2}(c + a_{n-1})\)
6. Simplifying the subtraction, assuming:
   • \(a_n - a_{n-1} = \frac{1}{2}(c + a_n) - a_{n-1}\)\)
7. Concluding: This gives the definition of an Arithmetic Progression (AP) where the difference between consecutive terms is constant.

Therefore, based on the derivation, the correct answer is that the sequence \(a_1, a_2, a_3, \dots\) is an Arithmetic Progression.