Analyzing the Number Series Pattern

The question asks us to find the number that replaces the question mark (?) in the given number series: 7, 7, 14, 42, ?, 840.

Let's examine the relationship between consecutive terms in the series to identify the underlying pattern.

• The first term is 7.
• The second term is 7. To get from the first term to the second, we can multiply by 1 ($7 \times 1 = 7$).
• The third term is 14. To get from the second term to the third, we can multiply by 2 ($7 \times 2 = 14$).
• The fourth term is 42. To get from the third term to the fourth, we can multiply by 3 ($14 \times 3 = 42$).

We can observe a pattern emerging: each term seems to be obtained by multiplying the previous term by a consecutively increasing integer, starting from 1.

Let $n_k$ represent the k-th term in the series. The pattern appears to be:

• $n_2 = n_1 \times 1$
• $n_3 = n_2 \times 2$
• $n_4 = n_3 \times 3$

Following this pattern, the next term, which is the fifth term (the position of the question mark), should be obtained by multiplying the fourth term by 4.

• Fifth term (?) = $n_4 \times 4 = 42 \times 4$

Let's calculate the value:

\( 42 \times 4 \)

We can break this down:

\( (40 + 2) \times 4 = (40 \times 4) + (2 \times 4) \)

\( 160 + 8 = 168 \)

So, the fifth term should be 168.

To confirm this pattern, let's check if multiplying the fifth term (168) by 5 gives the sixth term (840).

• Sixth term = $n_5 \times 5 = 168 \times 5$

\( 168 \times 5 \)

We can calculate this:

\( 168 \times 5 = 840 \)

This matches the last number in the given series (840). Therefore, the identified pattern is correct.

Step-by-Step Calculation of the Missing Number

Here is a summary of the steps to find the missing number in the series:

1. Identify the relationship between consecutive terms.
2. Notice that the multiplier increases by 1 for each step.
3. The pattern is $n_k = n_{k-1} \times (k-1)$ for $k \ge 2$. Let's adjust the index to make it simpler: the multiplier for the k-th term (from the (k-1)-th term) is $(k-1)$. Or, simpler still, the multiplier for the jump *after* the k-th term is k.
4. The jumps are: $\times 1$, $\times 2$, $\times 3$, $\times 4$, $\times 5$.
5. Apply the next multiplier in the sequence to the last known term before the question mark.
6. The last known term is 42 (the 4th term). The next multiplier is 4.
7. Calculate $42 \times 4$.
8. The result is 168.
9. Verify the pattern by checking the next step ($168 \times 5 = 840$).

Summary of the Number Series Pattern

The number that replaces the question mark is 168.

Revision Table: Number Series Patterns

Additional Information: Solving Number Series Questions

Solving number series questions requires careful observation and logical reasoning. Here are some tips:

• Look at the differences between consecutive terms. Are they constant? Do they form a pattern?
• Look at the ratios between consecutive terms. Are they constant? Do they form a pattern?
• Consider squares, cubes, square roots, or cube roots of numbers.
• Check for alternating patterns (e.g., a pattern applied to every other term).
• Look for patterns involving addition, subtraction, multiplication, division, or a combination of operations.
• Sometimes, the pattern involves two interleaved series.
• Practice with different types of series to become familiar with common patterns.

The series 7, 7, 14, 42, ?, 840 is an example of a series where the multiplier changes in a predictable way.