Question

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

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

Hint:

Check if each term relates to a power of 2, such as 2 raised to an increasing power minus 1, in addition to checking the relation between consecutive terms. Spotting the exponential form usually gives the next term faster than repeating the same calculation each time.

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 \( 127 \).