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Home » This Ceiling Function Will Change How You Solve Math Problems Forever! - AIKO, infinite ways to autonomy.

This Ceiling Function Will Change How You Solve Math Problems Forever! - AIKO, infinite ways to autonomy.

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 A geometric sequence has a first term of 3 and a common ratio of 2. Find the 6th term.

📌 Formula for \( n \)-th term: \( a_n = a_1 \times r^{n-1} \).

📌 \( a_6 = 3 \times 2^{6-1} = 3 \times 32 = 96 \).

📌 #### 96Question: What is the sum of all values of $ x $ for which $ \sqrt{(x - 3)^2} = 2x - 5 $?

📌 Solution: The equation $ \sqrt{(x - 3)^2} = 2x - 5 $ simplifies to $ |x - 3| = 2x - 5 $. This gives two cases:

📌 $ x - 3 = 2x - 5 $ → $ -3 + 5 = 2x - x $ → $ x = 2 $.

📌 $ -(x - 3) = 2x - 5 $ → $ -x + 3 = 2x - 5 $ → $ 3 + 5 = 3x $ → $ x = \frac{8}{3} $.

📌 Check validity: For $ x = 2 $, $ |2 - 3| = 1 $ and $ 2(2) - 5 = -1 $. Not valid since LHS ≠ RHS. For $ x = \frac{8}{3} $, $ \left|\frac{8}{3} - 3\right| = \frac{1}{3} $ and $ 2\left(\frac{8}{3}\right) - 5 = \frac{1}{3} $. Valid.

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