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Home » Each such pair yields an integer solution. Therefore, there are 16 lattice points. - Groen Casting

Each such pair yields an integer solution. Therefore, there are 16 lattice points. - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Find the sum of the series \( 1 + 3 + 5 + \dots + 99 \).

📌 This is an arithmetic series of odd numbers with first term \( a = 1 \), common difference \( d = 2 \), and last term \( l = 99 \).

📌 Number of terms \( n \): \( l = a + (n-1)d \rightarrow 99 = 1 + (n-1) \times 2 \).

📌 Solve: \( 98 = 2(n-1) \rightarrow n-1 = 49 \rightarrow n = 50 \).

📌 Sum of arithmetic series: \( S = \frac{n}{2}(a + l) \).

📌 \( S = \frac{50}{2}(1 + 99) = 25 \times 100 = 2500 \).

📌 A conical vessel with a base radius of 3 cm and height of 9 cm is filled with water. What is the volume of water in cubic centimeters?

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