Question

Complete the series: \(3, 7, 15, 31, 63, \ldots\)

• A. \(95\)
• B. \(111\)
• C. \(127\)
• D. \(128\)

Hint:

In number series problems, check simple operations such as multiplication, addition, subtraction, or powers between consecutive terms. Recognizing the pattern quickly helps determine the next number.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The goal is to identify the logical progression in the given sequence of numbers to find the next term.
Step 2: Key Formula or Approach:
Examine the difference between consecutive terms or look for a multiplier-addition pattern.
Pattern 1: \( \text{Term}_{n} \times 2 + 1 \).
Pattern 2: Difference increases as powers of \( 2 \) (\( 4, 8, 16, 32 \dots \)).
Step 3: Detailed Explanation:
Let's check the multiplier pattern:
\( 3 \times 2 + 1 = 7 \)
\( 7 \times 2 + 1 = 15 \)
\( 15 \times 2 + 1 = 31 \)
\( 31 \times 2 + 1 = 63 \)
The pattern is consistent. Following this for the next term:
\( 63 \times 2 + 1 = 126 + 1 = 127 \)
Alternatively, using differences:
\( 7 - 3 = 4 = 2^2 \)
\( 15 - 7 = 8 = 2^3 \)
\( 31 - 15 = 16 = 2^4 \)
\( 63 - 31 = 32 = 2^5 \)
The next difference should be \( 2^6 = 64 \).
\( 63 + 64 = 127 \)
Step 4: Final Answer:
The next number in the series is \( 127 \).