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

Let $a_1,a_2,a_3,a_4$ be an A.P. of four terms such that each term of the A.P. and its common difference are integers. If $a_1+a_2+a_3+a_4=48$ and $a_1^2a_2a_3a_4+1^4=361$, then the largest term of the A.P. is equal to

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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