DIRECTION for the question: In the following question, a series is given with term/s missing. Choose the correct alternative to replace the question mark in the given series.
5, 6, 9, 15, ?, 40
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In number series problems:
• First, always check the difference between consecutive terms.
• If the differences don't form a constant sequence, check if the differences themselves form an arithmetic progression (double difference).
Step 1: Understanding the Concept:
Analyze the differences between consecutive terms to identify a pattern (arithmetic or geometric progression). Step 2: Detailed Explanation:
Difference between 5 and 6 = 1
Difference between 6 and 9 = 3
Difference between 9 and 15 = 6
Difference between 15 and ? = $x$
Difference between ? and 40 = $y$ Step 3: Calculation:
The sequence of differences is 1, 3, 6... These are triangular numbers (sum of consecutive integers).
$1 (+2) \rightarrow 3 (+3) \rightarrow 6 (+4) \rightarrow 10 (+5) \rightarrow 15$.
So, the next difference should be 10.
$15 + 10 = 25$.
Verification: $25 + 15 = 40$. The pattern holds. Step 4: Final Answer:
The missing number is 25. Thus, the correct option is (b).