Finding the Next Term in the Number Series

The question asks us to identify the next term in the given number series: 4, 12, 16, 80, 86, 602, ?

To find the next term in the series, we need to analyze the pattern between consecutive numbers. Let's look at the relationship between each pair of terms:

• From 4 to 12: $12 = 4 \times 3$
• From 12 to 16: $16 = 12 + 4$
• From 16 to 80: $80 = 16 \times 5$
• From 80 to 86: $86 = 80 + 6$
• From 86 to 602: $602 = 86 \times 7$

Observing the operations and the numbers involved, we can see a repeating pattern:

1. The operations alternate between multiplication ($\times$) and addition ($+$).
2. The number used in the operation increases by 1 each time, starting from 3 (for multiplication), then 4 (for addition), then 5 (for multiplication), and so on.

Let's summarize the pattern discovered:

• Term 1 to Term 2: Multiply by 3
• Term 2 to Term 3: Add 4
• Term 3 to Term 4: Multiply by 5
• Term 4 to Term 5: Add 6
• Term 5 to Term 6: Multiply by 7

Following this established pattern, the next operation after multiplying by 7 should be adding the next number in the sequence 3, 4, 5, 6, 7, which is 8.

So, to find the 7th term (the term after 602), we should add 8 to the 6th term (602).

Calculation for the next term:

$602 + 8 = 610$

Therefore, the next term in the series is 610.

Understanding Number Series Patterns

Number series questions test logical reasoning and pattern recognition. Common patterns include arithmetic progressions, geometric progressions, differences between terms, ratios between terms, alternating operations, or a combination of these. Identifying the underlying rule is key to solving such problems.

Step-by-Step Solution Analysis

Let the series be $T_1, T_2, T_3, T_4, T_5, T_6, T_7$. The given series is 4, 12, 16, 80, 86, 602, ?

• $T_1 = 4$
• $T_2 = 12$. $T_2 = T_1 \times 3 = 4 \times 3 = 12$.
• $T_3 = 16$. $T_3 = T_2 + 4 = 12 + 4 = 16$.
• $T_4 = 80$. $T_4 = T_3 \times 5 = 16 \times 5 = 80$.
• $T_5 = 86$. $T_5 = T_4 + 6 = 80 + 6 = 86$.
• $T_6 = 602$. $T_6 = T_5 \times 7 = 86 \times 7 = 602$.
• $T_7 = ?$. Based on the pattern, the next operation is addition, and the number to add is 8. $T_7 = T_6 + 8 = 602 + 8 = 610$.

The pattern is: $\times 3, + 4, \times 5, + 6, \times 7, + 8, \dots$

Revision Table: Number Series Pattern

Additional Information: Types of Number Series

Understanding different types of series patterns can help in solving number series problems quickly. Some common types include:

• Arithmetic Series: A constant difference between consecutive terms (e.g., 2, 5, 8, 11... difference is 3).
• Geometric Series: A constant ratio between consecutive terms (e.g., 3, 6, 12, 24... ratio is 2).
• Difference Series: The difference between consecutive terms follows a pattern (e.g., 1, 3, 7, 13, 21... differences are 2, 4, 6, 8...).
• Alternating Series: Operations or values alternate between terms (like the series in this problem).
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5, 8...).
• Mixed Series: A combination of multiple patterns or operations.

Practicing with various types of series is crucial for improving pattern recognition skills.