A pattern of black and white squares follows a rule. In the sequence below, the number of black squares in successive figures are 1, 5, 13, 25, 41... What is the number of black squares in the 6th figure?

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EEJ MAIN Aptitude QUESTION #10930

Question 1

A pattern of black and white squares follows a rule. In the sequence below, the number of black squares in successive figures are 1, 5, 13, 25, 41... What is the number of black squares in the 6th figure?

57
61✔️
65
69

Correct Answer Explanation

Finding differences: \(5-1=4,\ 13-5=8,\ 25-13=12,\ 41-25=16\

Second differences: 4, 4, 4 — constant. So differences increase by 4 each time.

Next difference = 20. 6th term = \(41 + 20 = \mathbf{61}\

These represent centered square numbers: \(2n^2 - 2n + 1\ for \(n = 1, 2, 3...\

\(n=6: 2(36) - 12 + 1 = 72 - 12 + 1 = 61\ ✓

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