Let \( a_1, a_2, \dots, a_{2024} \) be an Arithmetic Progression such that \[ a_1 + (a_1 + a_0 + a_1 + a_2 + \cdots + a_{2020} + a_{2024}) = 2233. \quad \text{Then} \quad a_1 + a_2 + a_3 + \dots + a_{2022} \] is equal to ____ :

Question

For some positive and distinct real numbers \(x ,y\), and \(z\) , if \(\frac{1}{\sqrt{ y}+ \sqrt{z}}\) is the arithmetic mean of \(\frac{1}{\sqrt{x}+ \sqrt{z}}\) and \(\frac{1}{\sqrt{x} +\sqrt{y}}\) , then the relationship which will always hold true, is

Answer 1

Given the sum: \[ a_1 + a_2 + \dots + a_{2024} = 2233 \] In an Arithmetic Progression (A.P.), terms equidistant from the ends sum to the same value: \[ a_1 + a_{2024} = a_2 + a_{2023} = \dots = a_{1012} + a_{1013} \] This results in: \[ 203 \quad \text{pairs of the form} \quad (a_1 + a_{2024}) \] The sum of the first 2024 terms is calculated as: \[ S_{2024} = \frac{2024}{2} (a_1 + a_{2024}) = 2233 \] Using the A.P. sum formula: \[ S = 2024 \times 11 \] \(\text{Therefore, the final sum is:}\) \[ \boxed{11132} \]

Let \( a_1, a_2, \dots, a_{2024} \) be an Arithmetic Progression such that \[ a_1 + (a_1 + a_0 + a_1 + a_2 + \cdots + a_{2020} + a_{2024}) = 2233. \quad \text{Then} \quad a_1 + a_2 + a_3 + \dots + a_{2022} \] is equal to ____ :

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Correct Answer: 11132

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