Number Series Pattern Analysis

We are asked to find the missing number in the series: 1, 3, 7, 15, 31, 63, ?. Let's analyze the relationship between consecutive terms to identify the pattern.

Method 1: Analyzing Differences

First, let's calculate the difference between each term and the next one:

• $3 - 1 = 2$
• $7 - 3 = 4$
• $15 - 7 = 8$
• $31 - 15 = 16$
• $63 - 31 = 32$

The differences are 2, 4, 8, 16, 32. We can see that each difference is double the previous difference. This is a sequence of powers of 2:

• $2 = 2^1$
• $4 = 2^2$
• $8 = 2^3$
• $16 = 2^4$
• $32 = 2^5$

Following this pattern, the next difference should be $2^6$.

Method 2: Relation to Powers of 2

Let's examine the terms themselves in relation to powers of 2:

• $1 = 2^1 - 1$
• $3 = 2^2 - 1$
• $7 = 2^3 - 1$
• $15 = 2^4 - 1$
• $31 = 2^5 - 1$
• $63 = 2^6 - 1$

This shows a clear pattern where each term is one less than a power of 2, starting from $2^1$.

Method 3: Doubling and Adding

Another way to see the pattern is by doubling the current term and adding 1:

• $(1 \times 2) + 1 = 2 + 1 = 3$
• $(3 \times 2) + 1 = 6 + 1 = 7$
• $(7 \times 2) + 1 = 14 + 1 = 15$
• $(15 \times 2) + 1 = 30 + 1 = 31$
• $(31 \times 2) + 1 = 62 + 1 = 63$

This relationship holds true for all the given terms.

Calculating the Next Term

Using any of the identified patterns, we can find the next term in the series.

Using Method 1 (Differences):

The next difference is $2^6 = 64$. To find the next term, we add this difference to the last term:

$63 + 64 = 127$

Using Method 2 (Powers of 2):

The next term should be $2^7 - 1$.

$2^7 - 1 = 128 - 1 = 127$

Using Method 3 (Doubling and Adding):

We double the last term (63) and add 1:

$(63 \times 2) + 1 = 126 + 1 = 127$

All methods consistently show that the next number in the series is 127.

Final Answer Determination

The number series follows a pattern where each term is obtained by doubling the previous term and adding 1, or alternatively, each term is of the form $2^n - 1$. The next term in the sequence 1, 3, 7, 15, 31, 63, ? is calculated as follows:

Last term = 63

Next term = $(63 \times 2) + 1 = 126 + 1 = 127$.

Alternatively, using the formula $2^n - 1$, the terms correspond to $n=1, 2, 3, 4, 5, 6$. The next term corresponds to $n=7$.

Next term = $2^7 - 1 = 128 - 1 = 127$.

Therefore, the number that can replace the question mark (?) is 127.