Question:medium

Fill in the blank: 
257, 291, ______, 365, 405

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When two consecutive gaps are visible, test if differences themselves form an arithmetic progression.
Updated On: Jul 15, 2026
  • 313
  • 322
  • 327
  • 343
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The Correct Option is C

Approach Solution - 1

Step 1: Understanding the Concept:
Instead of only checking the gaps between terms, compare each term to nearby perfect squares to see if a cleaner pattern exists.

Step 2: Key Formula or Approach:
Test whether each term equals \((15+k)^2 + k\), where \(k\) is the term's position in the sequence (1 for the first term, 2 for the second, and so on).

Step 3: Detailed Explanation:
For \(k=1\): \(16^2 + 1 = 256 + 1 = 257\), which matches the first term.
For \(k=2\): \(17^2 + 2 = 289 + 2 = 291\), which matches the second term.
For \(k=4\): \(19^2 + 4 = 361 + 4 = 365\), which matches the fourth term.
For \(k=5\): \(20^2 + 5 = 400 + 5 = 405\), which matches the fifth term.
Since the formula holds for every known term, the missing third term (\(k=3\)) should equal \(18^2 + 3 = 324 + 3 = 327\).

Step 4: Final Answer:
The missing number is 327.
\[ \boxed{327} \]
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Approach Solution -2

Step 1: Understanding the Concept:
The gaps between consecutive terms often form their own arithmetic progression in these sequences, so instead of finding the missing term directly, find the missing gap.

Step 2: Key Formula or Approach:
Treat the gap between term 4 and term 5, and the gap between term 1 and term 2, as the first and fourth terms of an arithmetic progression of four gaps, \(D_1, D_2, D_3, D_4\), and use \(D_n = D_1 + (n-1)d\) to find the common difference \(d\).

Step 3: Detailed Explanation:
The first gap is \(D_1 = 291 - 257 = 34\), and the fourth gap is \(D_4 = 405 - 365 = 40\). Using \(D_4 = D_1 + 3d\), we get \(40 = 34 + 3d\), so \(3d = 6\) and \(d = 2\).
This gives \(D_2 = D_1 + d = 34 + 2 = 36\) and \(D_3 = D_2 + d = 36 + 2 = 38\). Checking, \(D_3\) should also equal \(D_4 - d = 40 - 2 = 38\), which matches.
The missing term equals the second term, which is \(257 + D_1 + D_2 = 257 + 34 + 36 = 327\).

Step 4: Final Answer:
The missing number is 327.
\[ \boxed{327} \]
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