Question:
Fill in the blank :
257, 291, ______ , 365, 405
Fill in the blank :
257, 291, ______ , 365, 405
257, 291, ______ , 365, 405
Updated On: Jul 15, 2026
- 313
- 322
- 327
- 343
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The Correct Option is C
Approach Solution - 1
The correct option is (C): 327.
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Approach Solution -2
Compute the consecutive differences of the known terms: \(291-257=34\) and \(405-365=40\). Since the differences of consecutive terms rise by 2 each step (34, 36, 38, 40), the gap between 291 and the missing term should be 36, and the gap between the missing term and 365 should be 38.
- Option A (313): \(313-291=22\) and \(365-313=52\), neither matches the expected 36/38 gap.
- Option B (322): \(322-291=31\) and \(365-322=43\), still doesn't match 36/38.
- Option C (327): \(327-291=36\) and \(365-327=38\), both differences land exactly on the expected 36 and 38.
- Option D (343): \(343-291=52\) and \(365-343=22\), far off the expected gaps.
Only option C keeps the second-difference pattern rising smoothly by 2 throughout the series.
Therefore, the correct answer is 327.
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Approach Solution -3
Since the first differences of the series, \(291-257=34\) and \(405-365=40\), rise by a constant 2 each step (34, 36, 38, 40), the whole sequence follows a quadratic formula. Writing the n-th term as \(u(n) = n^2 + 31n + 225\) reproduces every given value: \(u(1)=1+31+225=257\), \(u(2)=4+62+225=291\), \(u(4)=16+124+225=365\), \(u(5)=25+155+225=405\), all matching the series exactly, so the same formula gives the missing third term directly: \(u(3)=9+93+225=327\).
- Option A (313): the formula gives 327 for \(n=3\), not 313, so this value does not fit the quadratic that generates the rest of the series.
- Option B (322): also falls short of the formula's output of 327 for \(n=3\), so it breaks the pattern the other four terms follow.
- Option C (327): matches \(u(3)=9+93+225=327\) exactly, the same formula that correctly reproduces all four given terms.
- Option D (343): is well above the formula's value of 327 for \(n=3\), overshooting the pattern set by the rest of the series.
The quadratic formula fitted from the four known terms confirms the missing term is 327.
Therefore, the correct answer is 327.
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