Understanding Number Series Patterns

A number series is a sequence of numbers that follows a specific pattern. To solve problems involving number series, we need to identify this pattern, which could involve addition, subtraction, multiplication, division, squares, cubes, or a combination of these operations applied to consecutive terms or their positions in the series.

Analyzing the Given Number Series

The given series is: 5, 8, 13, 20, 29, ........., 53, 68, 85

Let's examine the differences between consecutive terms to find the pattern:

• Difference between the 2nd and 1st term: $\small 8 - 5 = 3$
• Difference between the 3rd and 2nd term: $\small 13 - 8 = 5$
• Difference between the 4th and 3rd term: $\small 20 - 13 = 7$
• Difference between the 5th and 4th term: $\small 29 - 20 = 9$

We can see that the differences are 3, 5, 7, 9. This sequence of differences itself follows a pattern: it's an arithmetic progression where each term increases by 2.

• $\small 5 - 3 = 2$
• $\small 7 - 5 = 2$
• $\small 9 - 7 = 2$

So, the difference between consecutive terms is increasing by 2 at each step.

Calculating the Missing Term

Following this pattern of differences:

• The next difference after 9 should be $\small 9 + 2 = 11$.
• To find the missing term, we add this difference (11) to the last known term (29).
• Missing term = $\small 29 + 11 = 40$.

Verifying the Pattern with Subsequent Terms

Let's check if the subsequent terms in the series follow the same pattern with the calculated missing term (40).

• The next difference after 11 should be $\small 11 + 2 = 13$.
• The term after 40 should be $\small 40 + 13 = 53$. (This matches the given series)
• The next difference after 13 should be $\small 13 + 2 = 15$.
• The term after 53 should be $\small 53 + 15 = 68$. (This matches the given series)
• The next difference after 15 should be $\small 15 + 2 = 17$.
• The term after 68 should be $\small 68 + 17 = 85$. (This matches the given series)

The pattern holds true throughout the series with the missing term being 40.

Summary of the Series and Differences

The differences (3, 5, 7, 9, 11, 13, 15, 17) form an arithmetic progression with a common difference of 2. The missing term is indeed 40.

Revision Table: Number Series Pattern

Additional Information on Number Series

Number series problems are common in aptitude tests. They test your ability to identify logical patterns. Here are some common types of patterns you might encounter:

• Arithmetic Series: Each term is obtained by adding a constant value to the previous term (e.g., 2, 4, 6, 8...). The differences between consecutive terms are constant.
• Geometric Series: Each term is obtained by multiplying the previous term by a constant value (e.g., 3, 6, 12, 24...). The ratios between consecutive terms are constant.
• Series with Increasing/Decreasing Differences: Like the example solved, the differences between consecutive terms themselves form an arithmetic series (e.g., differences are +2, +4, +6, ... or -3, -6, -9, ...).
• Fibonacci Series: Each term is the sum of the two preceding terms (starting usually from 0 and 1, or 1 and 1) (e.g., 0, 1, 1, 2, 3, 5, 8...).
• Prime Number Series: The series consists of prime numbers in order (e.g., 2, 3, 5, 7, 11...).
• Square or Cube Series: Terms are squares or cubes of natural numbers, or variations involving squares/cubes (e.g., 1, 4, 9, 16... or 1, 8, 27, 64... or $\small n^2 \pm k$ or $\small n^3 \pm k$).
• Mixed Series: Combinations of different patterns (e.g., alternating arithmetic and geometric operations, or two intermingled series).

Solving number series requires careful observation and testing different possible patterns.