Question

Find the missing number in the series: 3, 7, 15, 31, 63, __?

• A. 95
• B. 111
• C. 127
• D. 135

Hint:

In number series problems, always check for common patterns such as:

• Multiplication followed by addition/subtraction
• Squares or cubes of numbers
• Powers of 2 or 3
• Increasing differences between terms

For this series: \[ 3, 7, 15, 31, 63 \] The pattern is: \[ 2^2-1,\; 2^3-1,\; 2^4-1,\; 2^5-1,\; 2^6-1 \] So the next term is: \[ 2^7 - 1 = 127 \]

Answer 1

The Correct Option is C

Concept:
Number series questions often follow a hidden pattern based on arithmetic operations such as addition, subtraction, multiplication, powers, or combinations of these operations. One common pattern is a sequence where each term is obtained by multiplying the previous term by a constant and then adding or subtracting a fixed number. Another useful way to analyze number series is to examine:

• Differences between consecutive terms
• Multiplicative relationships
• Patterns involving powers of numbers

Step 1: Observe the pattern in the given series.
\[ 3,\; 7,\; 15,\; 31,\; 63 \] Let us check if each term follows a multiplication pattern. \[ 3 \times 2 + 1 = 7 \] \[ 7 \times 2 + 1 = 15 \] \[ 15 \times 2 + 1 = 31 \] \[ 31 \times 2 + 1 = 63 \] Thus, the rule is: \[ \text{Next term} = (\text{Previous term} \times 2) + 1 \]
Step 2: Apply the pattern to find the next term.
\[ 63 \times 2 + 1 = 126 + 1 = 127 \]

Step 3: Verify the pattern using another perspective.
The sequence can also be written as: \[ 3 = 2^2 - 1 \] \[ 7 = 2^3 - 1 \] \[ 15 = 2^4 - 1 \] \[ 31 = 2^5 - 1 \] \[ 63 = 2^6 - 1 \] Therefore, the next term should be: \[ 2^7 - 1 = 128 - 1 = 127 \]
Step 4: Selecting the correct answer.
\[ \boxed{127} \] Thus, the missing number in the series is 127.