Question:medium

Examine the following numbers and identify the next number:
53, 50, 45, 38, 29, _____

Updated On: Jul 15, 2026
  • 11
  • 17
  • 18
  • 21
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The Correct Option is C

Approach Solution - 1

Step 1: Compute the differences between consecutive terms: \(50-53=-3\), \(45-50=-5\), \(38-45=-7\), \(29-38=-9\).

Step 2: Notice these differences are consecutive odd numbers decreasing by 2 each time, -3, -5, -7, -9, so the next difference is -11.

Step 3: Apply it to the last term: \(29+(-11)=18\).
\[ \boxed{18} \]
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Approach Solution -2

Since the differences between consecutive terms shrink by a constant amount each time, the terms fit a quadratic rule of the form \( T(n) = an^2+bn+c \), where \( n \) is the position in the sequence, 1, 2, 3, and so on. Using the first two terms, \( T(1)=53 \) and \( T(2)=50 \), together with the fact that the second difference is \( -2 \) (so \( a=-1 \)), gives \( b=0 \) and \( c=54 \), so \( T(n)=54-n^2 \). Checking, \( T(3)=54-9=45 \), \( T(4)=54-16=38 \), \( T(5)=54-25=29 \), all correct. The 6th term is \( T(6)=54-36=18 \).

  1. Option A (11): would need \( 54-n^2=11 \), giving \( n^2=43 \), which is not a perfect square, so it does not fit the position-based rule.
  2. Option B (17): would need \( n^2=37 \), also not a perfect square, so it does not fit either.
  3. Option C (18): gives \( n^2=36 \), exactly \( 6^2 \), matching position \( n=6 \) precisely.
  4. Option D (21): would need \( n^2=33 \), not a perfect square, so it does not fit the rule.

Only 18 satisfies \( T(n)=54-n^2 \) at the required 6th position.

Therefore, the correct answer is 18.

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