Question:medium

20, 30, 42, 56, 72, ____

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If first differences form their own simple AP, extend that AP rather than guessing the next term directly.
Updated On: Jul 15, 2026
  • 82
  • 90
  • 92
  • 94
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The Correct Option is B

Approach Solution - 1

Step 1: Understanding the Concept:
Check whether each term can be written as the product of two consecutive integers instead of only tracking the gaps between terms.

Step 2: Key Formula or Approach:
Test the formula \(k \times (k+1)\), where \(k\) starts at 4 for the first term and increases by 1 for each following term.

Step 3: Detailed Explanation:
For \(k=4\): \(4 \times 5 = 20\), matches the first term.
For \(k=5\): \(5 \times 6 = 30\), matches the second term.
For \(k=6\): \(6 \times 7 = 42\), matches the third term.
For \(k=7\): \(7 \times 8 = 56\), matches the fourth term.
For \(k=8\): \(8 \times 9 = 72\), matches the fifth term.
Since this formula holds for every given term, the sixth term uses \(k=9\): \(9 \times 10 = 90\).

Step 4: Final Answer:
The missing number is 90.
\[ \boxed{90} \]
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Approach Solution -2

Step 1: Understanding the Concept:
The gaps between consecutive terms, 10, 12, 14, 16, increase by a constant 2 each time, so the gap between term \(k\) and term \(k+1\) can be written as a general formula, \(8+2k\), instead of extending the list one step at a time.

Step 2: Key Formula or Approach:
Use \(8+2k\) for the gap after the \(k\)-th term, and plug in \(k=5\) directly to get the gap between the fifth and sixth terms.

Step 3: Detailed Explanation:
Checking the formula: for \(k=1\), \(8+2=10\), matching the rise from 20 to 30. For \(k=2\), \(8+4=12\), matching the rise from 30 to 42. For \(k=3\), \(8+6=14\), matching the rise from 42 to 56. For \(k=4\), \(8+8=16\), matching the rise from 56 to 72.
For the gap between the fifth and sixth terms, \(k=5\): \(8+10=18\).
So the sixth term is \(72+18=90\).

Step 4: Final Answer:
The missing number is 90.
\[ \boxed{90} \]
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