This is a number series problem where we need to identify the pattern governing the sequence of numbers to find the next term. Step 1: Understanding the Question:
The task is to find the number that logically follows 63 in the given series. Step 2: Key Formula or Approach:
We can analyze the series in two common ways:
1. Find the relationship between consecutive terms.
2. Check if the terms follow a pattern related to powers of a number. Step 3: Detailed Explanation: Method 1: Relationship between consecutive terms
Let's see how each term is generated from the previous one.
\[ 3 \rightarrow 7: \quad (3 \times 2) + 1 = 6 + 1 = 7 \]
\[ 7 \rightarrow 15: \quad (7 \times 2) + 1 = 14 + 1 = 15 \]
\[ 15 \rightarrow 31: \quad (15 \times 2) + 1 = 30 + 1 = 31 \]
\[ 31 \rightarrow 63: \quad (31 \times 2) + 1 = 62 + 1 = 63 \]
The pattern is consistent: Next Term = (Previous Term × 2) + 1.
Applying this rule to find the missing number:
\[ (63 \times 2) + 1 = 126 + 1 = 127 \]
Method 2: Pattern based on powers of 2
Let's examine each term in relation to powers of 2.
\[ 3 = 4 - 1 = 2^2 - 1 \]
\[ 7 = 8 - 1 = 2^3 - 1 \]
\[ 15 = 16 - 1 = 2^4 - 1 \]
\[ 31 = 32 - 1 = 2^5 - 1 \]
\[ 63 = 64 - 1 = 2^6 - 1 \]
The pattern is: The \(n\)-th term is \(2^{n+1} - 1\). The last given term (63) is the 5th term in the sequence, which is \(2^{5+1} - 1 = 2^6 - 1\).
The next term (the 6th term) would be \(2^{6+1} - 1\):
\[ 2^7 - 1 = 128 - 1 = 127 \]
Both methods yield the same result. Step 4: Final Answer:
The missing number in the series is 127.
\[
\boxed{127}
\]