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\( 2024 ( 2025 ^ { 2024 } + 2025 ^ { 2023 } + 2025 ^ { 2022 } + \ldots + 1 ) + 1 \)

Ask by Li Maxwell. in Cambodia
Jan 10,2025

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Answer

The simplified form of the expression is \(2025^{2025}\).

Solution

Let's simplify the given expression: \[ 2024 \left( 2025^{2024} + 2025^{2023} + 2025^{2022} + \ldots + 1 \right) + 1 \] ### Step 1: Recognize the Geometric Series The expression inside the parentheses is a finite geometric series: \[ S = 2025^{2024} + 2025^{2023} + 2025^{2022} + \ldots + 1 \] This series has: - **First term (a):** \(1\) - **Common ratio (r):** \(2025\) - **Number of terms (n):** \(2025\) (from \(2025^0\) to \(2025^{2024}\)) ### Step 2: Sum of the Geometric Series The sum of a finite geometric series is given by: \[ S = \frac{r^n - 1}{r - 1} \] Plugging in the values: \[ S = \frac{2025^{2025} - 1}{2025 - 1} = \frac{2025^{2025} - 1}{2024} \] ### Step 3: Multiply by 2024 and Add 1 Substitute the sum back into the original expression: \[ 2024 \times S + 1 = 2024 \times \left( \frac{2025^{2025} - 1}{2024} \right) + 1 \] Simplify: \[ = 2025^{2025} - 1 + 1 = 2025^{2025} \] ### Final Simplified Form \[ 2024 \left( 2025^{2024} + 2025^{2023} + 2025^{2022} + \ldots + 1 \right) + 1 = 2025^{2025} \] **Therefore, the simplified form of the given expression is:** \[ 2025^{2025} \]

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To simplify the expression \( 2024 ( 2025 ^ { 2024 } + 2025 ^ { 2023 } + 2025 ^ { 2022 } + \ldots + 1 ) + 1 \), we start by recognizing that the inside of the parentheses is a geometric series. The geometric series can be summed up using the formula: \[ S_n = a \cdot \frac{r^n - 1}{r - 1} \] where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. In our series, the first term \( a = 1 \), the common ratio \( r = 2025 \), and the number of terms is \( 2024 \) (from \( 2025^0 \) to \( 2025^{2024} \)). So, the sum of the series is: \[ S = 1 \cdot \frac{2025^{2024} - 1}{2025 - 1} \] This simplifies down to: \[ S = \frac{2025^{2024} - 1}{2024} \] Substituting this back into our original expression gives: \[ 2024 \cdot S + 1 = 2024 \cdot \frac{2025^{2024} - 1}{2024} + 1 \] The \( 2024 \) in the numerator and denominator cancels out: \[ = 2025^{2024} - 1 + 1 \] This simplifies further to: \[ = 2025^{2024} \] Thus, the final result is: \[ \boxed{2025^{2024}} \]

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