Question:
Fill in the blank:
257, 291, ______, 365, 405
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
We need to identify the missing number in the sequence: 257, 291, ______, 365, 405.
The best approach is to first determine the pattern or rule governing the sequence. Let's calculate the difference between consecutive terms where possible:
- From 257 to 291: 291 - 257 = 34
- From 365 to 405: 405 - 365 = 40
We hypothesize that the differences might increment by a constant value. The difference might form an arithmetic sequence. To test this, we calculate the difference increase from 34 to 40:
- Increase from 34 to 40: 40 - 34 = 6
Therefore, we suspect the missing difference between 291 and the missing middle number could be 34 + 6 = 40.
Using this logic:
- From 291 to the missing number: 291 + 36 = 327
Thus, the missing number that fits the sequence is 327, as the differences would become 34, 36, 38, and 40, showing a steady increment of 2 in the difference values.
Therefore, the correct answer is: 327
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Approach Solution -2
Instead of tracking gaps between consecutive terms, notice that each number in this sequence is close to a perfect square. Checking against squares of consecutive integers starting at 16 quickly reveals the pattern: \(16^2 = 256\), and the first term is 257, just 1 more. \(17^2 = 289\), and the second term is 291, 2 more. \(19^2 = 361\), and the fourth term is 365, 4 more. \(20^2 = 400\), and the fifth term is 405, 5 more. So each term equals \((15+k)^2 + k\), where \(k\) is the position of the term in the sequence (1, 2, 3, 4, 5).
For the missing third term, \(k = 3\), so the value should be \(18^2 + 3 = 324 + 3 = 327\).
- 313: This does not equal \(18^2 + 3\); it is 14 short of the value the pattern predicts, so it breaks the square-plus-position rule.
- 322: This is 5 short of \(18^2 + 3 = 327\), and does not fit the consistent offset pattern of \(+1, +2, +3, +4, +5\) seen in the rest of the sequence.
- 327: This equals \(18^2 + 3 = 324 + 3\), exactly continuing the offset pattern of \(+1, +2, +3, +4, +5\) that all the other terms follow.
- 343: This overshoots the predicted value by 16 and does not correspond to any nearby perfect square plus a small offset, so it breaks the pattern.
Since every other term in the sequence equals a perfect square (16 squared through 20 squared) plus its position number, the missing term must equal 18 squared plus 3, which is 327.
Therefore, the correct answer is 327.
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Approach Solution -3
Since the sequence looks like it could follow a quadratic rule rather than a simple linear one, we can model each term as \( T_k = Ak^2 + Bk + C \), where \(k\) is the term's position (1 for 257, 2 for the missing term, 4 for 365, and 5 for 405), and solve for the constants \(A\), \(B\), and \(C\) using the terms we already know.
Using position 1: \(A + B + C = 257\). Using position 4: \(16A + 4B + C = 365\). Using position 5: \(25A + 5B + C = 405\). Subtracting the first equation from the equation at position 4 gives \(15A + 3B = 108\), and subtracting the equation at position 4 from the equation at position 5 gives \(9A + B = 40\). Solving these two together, from the second, \(B = 40 - 9A\), substituting into the first, \(15A + 3(40 - 9A) = 108\), which simplifies to \(15A + 120 - 27A = 108\), so \(-12A = -12\), giving \(A = 1\). Then \(B = 40 - 9 = 31\), and from the first original equation, \(C = 257 - 1 - 31 = 225\).
So the rule is \( T_k = k^2 + 31k + 225 \). Checking it against position 2 for the second term (already known to be 291): \(4 + 62 + 225 = 291\), which matches, confirming the formula is correct. The missing term is at position 3: \(T_3 = 9 + 93 + 225 = 327\).
- 313: Substituting into \(T_k = k^2 + 31k + 225\) at \(k=3\) gives 327, not 313, so this value does not satisfy the quadratic rule fitted from the other four terms.
- 322: This is 5 short of the value the formula predicts for position 3, so it does not solve the equation \(k^2+31k+225=327\).
- 327: This exactly matches \(T_3 = 9 + 93 + 225 = 327\), confirming the quadratic model fitted from the other four terms.
- 343: This overshoots the formula's prediction by 16 and does not fit \(T_k = k^2+31k+225\) at \(k=3\).
Solving the quadratic model built from the four known terms confirms the missing term is 327.
Therefore, the correct answer is 327.
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