Question:medium

Fill in the blank :
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

Step 1: Find the differences between the known consecutive terms: \(291-257=34\) and \(405-365=40\).

Step 2: Notice the gap between differences grows by 2 each time, so the missing differences must be 36 (between 291 and the blank) and 38 (between the blank and 365).

Step 3: Add the first of these to 291: \(291+36=327\), and verify \(327+38=365\), which matches the given next term.
\[ \boxed{327} \]
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Approach Solution -2

For a series whose first differences increase by a constant amount, the second difference of the series itself is fixed, here it equals 2, since the first differences climb by 2 each step. This means, for any three consecutive terms, the middle one satisfies \(u(\text{before}) - 2 \times u(\text{middle}) + u(\text{after}) = 2\). Applying this to the terms surrounding the blank, 291, the blank, and 365, gives \(291 - 2 \times u(\text{blank}) + 365 = 2\), so \(2 \times u(\text{blank}) = 654\) and \(u(\text{blank}) = 327\).

  1. Option A (313): substituting 313 gives \(291 - 2(313) + 365 = 30\), far from the required constant of 2.
  2. Option B (322): substituting 322 gives \(291-2(322)+365 = 12\), still not equal to 2.
  3. Option C (327): substituting 327 gives \(291 - 2(327) + 365 = 2\), exactly matching the constant second difference required.
  4. Option D (343): substituting 343 gives \(291-2(343)+365=-30\), far below the required value of 2.

Only 327 keeps the second difference of the series constant at 2 across the gap.

Therefore, the correct answer is 327.

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