Solving the Number Series: 2, 5, 11, 23, ?

Let's analyze the given number series to find the pattern and determine the missing number. The series is 2, 5, 11, 23, ?

We can look at the differences between consecutive terms:

• Difference between the 2nd and 1st term: ${5 - 2 = 3}$
• Difference between the 3rd and 2nd term: ${11 - 5 = 6}$
• Difference between the 4th and 3rd term: ${23 - 11 = 12}$

Observing the differences (3, 6, 12), we can see that each difference is double the previous difference:

• ${6 = 3 \times 2}$
• ${12 = 6 \times 2}$

This suggests a pattern where the difference between consecutive terms is doubling. To find the next term, the difference between the 5th and 4th term should be double the difference between the 4th and 3rd term (12). So, the next difference is ${12 \times 2 = 24}$.

Adding this difference to the last term (23) gives the missing number:

Missing number ${= 23 + 24 = 47}$.

Alternatively, we can look for a direct relationship between each term and the next. Let ${T_n}$ be the n-th term in the series:

• ${T_1 = 2}$
• ${T_2 = 5}$
• ${T_3 = 11}$
• ${T_4 = 23}$

Let's see if there's a formula linking ${T_n}$ to ${T_{n+1}}$:

• ${2 \times 2 + 1 = 4 + 1 = 5}$ (Matches ${T_2}$)
• ${5 \times 2 + 1 = 10 + 1 = 11}$ (Matches ${T_3}$)
• ${11 \times 2 + 1 = 22 + 1 = 23}$ (Matches ${T_4}$)

The pattern appears to be ${T_{n+1} = 2 \times T_n + 1}$. Let's apply this formula to find the next term after 23 (${T_4}$):

Missing number ${= 2 \times T_4 + 1 = 2 \times 23 + 1 = 46 + 1 = 47}$.

Both methods confirm that the missing number in the series is 47.

Revision Table: Number Series Pattern

Additional Information: Understanding Number Series

Number series questions are common in aptitude tests. They test your ability to identify patterns in a sequence of numbers. Common types of patterns include:

• Arithmetic Progression: A constant difference between consecutive terms (e.g., 2, 4, 6, 8...).
• Geometric Progression: A constant ratio between consecutive terms (e.g., 2, 4, 8, 16...).
• Difference Series: The differences between consecutive terms follow a pattern (as seen in this question, where differences form a geometric progression).
• Mixed Series: A combination of two or more patterns.
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5...).
• Square or Cube Series: Terms are squares or cubes of natural numbers, or involve operations on them.

To solve number series problems, try calculating differences, ratios, or looking for squares, cubes, or other common sequences. Sometimes, the pattern might relate terms that are not immediately adjacent.