Question

If \( \sum_{k=1}^{n} a_k = \alpha n^2 + \beta n \) and \( a_{10} = 59,\; a_6 = 7a_1 \), then find \( \alpha + \beta \):

• A. 5
• B. 10
• C. 8
• D. 3

Hint:

To find the nth term from a given sum formula, always use \( a_n = S_n - S_{n-1} \).

Answer 1

The Correct Option is A

Step 1: Find the general term \(a_n\).
Given \[ \sum_{k=1}^{n} a_k = \alpha n^2 + \beta n \] Then, \[ a_n = S_n - S_{n-1} \] \[ a_n = \alpha(2n-1) + \beta \]
Step 2: Use the given conditions.
For \( n = 10 \): \[ a_{10} = \alpha(19) + \beta = 59 \] For \( n = 6 \): \[ a_6 = \alpha(11) + \beta \] Given \( a_6 = 7a_1 \), where \[ a_1 = \alpha + \beta \] So, \[ \alpha(11) + \beta = 7(\alpha + \beta) \]
Step 3: Solve the equations.
From equations: \[ 19\alpha + \beta = 59 \] \[ 11\alpha + \beta = 7\alpha + 7\beta \] Solving, we get \[ \alpha = 2,\; \beta = 3 \]
Step 4: Final Answer.
\[ \alpha + \beta = 5 \]