Question

Which is the missing number in the given number sequence?
7, 15, 32, -----, 138, 281

• A. 67
• B. 65
• C. 69
• D. 63

Hint:

To find the pattern of rapidly increasing series, look at the ratio of consecutive terms. Since \( 15/7 \approx 2 \), \( 32/15 \approx 2 \), and \( 281/138 \approx 2 \), the pattern is highly likely to involve multiplication by 2.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find the missing term in the given numerical sequence by determining the underlying mathematical pattern.

Step 2: Detailed Explanation:
Let the terms of the sequence be \( T_1, T_2, T_3, T_4, T_5, T_6 \).
- \( T_1 = 7 \)
- \( T_2 = 15 \)
- \( T_3 = 32 \)
- \( T_4 = ? \)
- \( T_5 = 138 \)
- \( T_6 = 281 \)
Let us test a multiplication-addition pattern where each term is twice the preceding term plus an increasing integer:
\[ T_{n} = 2 \times T_{n-1} + (n - 1) \] Let us check:
- \( T_2 = 2 \times T_1 + 1 = 2 \times 7 + 1 = 15 \) (Correct)
- \( T_3 = 2 \times T_2 + 2 = 2 \times 15 + 2 = 32 \) (Correct)
Using this pattern, the fourth term \( T_4 \) should be:
- \( T_4 = 2 \times T_3 + 3 = 2 \times 32 + 3 = 64 + 3 = 67 \)
Let us verify if \( T_4 = 67 \) satisfies the rest of the sequence:
- \( T_5 = 2 \times T_4 + 4 = 2 \times 67 + 4 = 134 + 4 = 138 \) (Correct)
- \( T_6 = 2 \times T_5 + 5 = 2 \times 138 + 5 = 276 + 5 = 281 \) (Correct)
Since the pattern holds true, the missing number is 67.

Step 3: Final Answer:
(A) 67