Question

Fill in the missing number in the series:
2,6,15,_____, 56,92

• A. 30
• B. 32
• C. 34
• D. 31

Hint:

When the numbers in a series increase at an accelerating rate, always check the differences between consecutive terms.
If the differences are perfect squares ($4, 9, 16, 25, 36$), you can quickly identify the missing term.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To find the missing number in a mathematical series, we must analyze the differences between consecutive terms to identify the underlying pattern.

Step 2: Detailed Explanation:
Let us write down the given terms of the series:
\[ T_1 = 2, \quad T_2 = 6, \quad T_3 = 15, \quad T_4 = ?, \quad T_5 = 56, \quad T_6 = 92 \]
Let us calculate the differences between the known consecutive terms:
- First difference:
\[ T_2 - T_1 = 6 - 2 = 4 \]
Notice that \( 4 = 2^2 \).
- Second difference:
\[ T_3 - T_2 = 15 - 6 = 9 \]
Notice that \( 9 = 3^2 \).
The differences between consecutive terms appear to be the squares of consecutive integers: \( 2^2, 3^2, 4^2, 5^2, \dots \)
Let us test this pattern to find the missing fourth term (\( T_4 \)):
- If the pattern holds, the next difference should be \( 4^2 = 16 \):
\[ T_4 = T_3 + 16 = 15 + 16 = 31 \]
Let us verify if this value of \( T_4 = 31 \) satisfies the rest of the series:
- The next difference between \( T_4 \) and \( T_5 \) should be \( 5^2 = 25 \):
\[ T_5 = T_4 + 25 = 31 + 25 = 56 \]
This matches the fifth term given in the question ($56$).
- The final difference between \( T_5 \) and \( T_6 \) should be \( 6^2 = 36 \):
\[ T_6 = T_5 + 36 = 56 + 36 = 92 \]
This matches the sixth term given in the question ($92$).
Since the pattern is consistent across the entire series, the missing number is indeed $31$.

Step 3: Final Answer:
The missing number in the series is 31.