Solving the Number Series Question

The question asks us to find the number that replaces the question mark (?) in the given number series: 5, 17, 53, ?, 485.

To find the missing number, we need to identify the pattern or rule that governs the sequence of numbers in the series.

Identifying the Number Series Pattern

Let's examine the relationship between consecutive terms in the series.

• The first term is 5.
• The second term is 17.
• The third term is 53.
• The fourth term is the missing number (?).
• The fifth term is 485.

Let's look at the differences between consecutive terms:

• Difference between the 2nd and 1st term: $17 - 5 = 12$
• Difference between the 3rd and 2nd term: $53 - 17 = 36$

The differences are 12 and 36. We can observe that the second difference (36) is 3 times the first difference (12). Let's see if this pattern of multiplying the difference by 3 continues.

• If the next difference follows this pattern, it should be $36 \times 3 = 108$.
• The missing number is the third term plus this difference: $53 + 108 = 161$.

So, the fourth term could be 161. Let's check if the pattern holds for the next term.

• The difference between the fifth term and the calculated fourth term should be the previous difference (108) multiplied by 3: $108 \times 3 = 324$.
• Let's check if $485 - 161 = 324$. Yes, $485 - 161 = 324$.

This confirms the pattern of the differences being multiplied by 3 at each step.

Another Way to Look at the Pattern

Let's examine a direct relationship between consecutive terms:

• From 5 to 17: $5 \times 3 + 2 = 15 + 2 = 17$
• From 17 to 53: $17 \times 3 + 2 = 51 + 2 = 53$

This suggests a pattern where each term is obtained by multiplying the previous term by 3 and adding 2.

Let $T_n$ be the nth term in the series. The pattern seems to be $T_n = T_{n-1} \times 3 + 2$.

Let's use this rule to find the missing fourth term ($T_4$):

• $T_1 = 5$
• $T_2 = 17 = T_1 \times 3 + 2 = 5 \times 3 + 2$
• $T_3 = 53 = T_2 \times 3 + 2 = 17 \times 3 + 2$
• $T_4 = ? = T_3 \times 3 + 2 = 53 \times 3 + 2$

Calculating the fourth term:

$T_4 = 53 \times 3 + 2 = 159 + 2 = 161$

So, the missing number is 161.

Let's verify this with the fifth term ($T_5$):

• $T_5 = 485$. Using the rule, $T_5$ should be $T_4 \times 3 + 2$.
• $161 \times 3 + 2 = 483 + 2 = 485$.

This confirms that the pattern $T_n = T_{n-1} \times 3 + 2$ correctly describes the number series.

The Missing Number

Based on the identified pattern, the number that replaces the question mark (?) is 161.

Revision Table: Number Series Pattern

Additional Information: Solving Number Series

Solving number series problems involves finding the rule or pattern that connects the numbers in the sequence. Common patterns include:

• Arithmetic progression (constant difference)
• Geometric progression (constant ratio)
• Differences increasing or decreasing by a constant value or a multiple
• Alternating patterns
• Patterns involving squares, cubes, or other mathematical operations
• Combination of different arithmetic operations

It is often helpful to look at the differences between consecutive terms first. If the differences don't follow a simple pattern, look at the differences of the differences (second-order differences), or try to find a direct relationship between a term and its preceding term(s).