Analyzing the Given Number Series

The number series provided is 13, 10, 7, 4, ______.

We need to identify the pattern or rule that governs the progression of the numbers in this sequence to find the next term.

Identifying the Pattern in the Number Sequence

Let's examine the difference between consecutive terms in the series:

• Difference between the second term (10) and the first term (13): $10 - 13 = -3$
• Difference between the third term (7) and the second term (10): $7 - 10 = -3$
• Difference between the fourth term (4) and the third term (7): $4 - 7 = -3$

It is clear that there is a constant difference of -3 between each successive term. This indicates that the series is an arithmetic progression where each term is obtained by subtracting 3 from the previous term.

Calculating the Next Term in the Series

To find the next term in the series, we must apply the identified pattern. We take the last given term, which is 4, and subtract 3 from it.

Calculation:

Next term = Last term - Constant difference

Next term = $4 - 3$

Next term = $1$

Conclusion: Completing the Number Series

Following the pattern of subtracting 3 from the preceding term, the next number in the series 13, 10, 7, 4, ______ is 1.

The completed series is 13, 10, 7, 4, 1.

Revision Table: Number Series Analysis

Additional Information: Arithmetic Progressions Explained

A number series like this, where the difference between consecutive terms is constant, is called an arithmetic progression (AP). The constant difference is known as the common difference, often denoted by 'd'.

In this specific series: 13, 10, 7, 4, 1...

• The first term (a₁) is 13.
• The common difference (d) is -3.

The formula for the n-th term of an arithmetic progression is given by:

$a_n = a_1 + (n-1)d$

Using this formula, let's verify the terms:

• For n=1: $a_1 = 13 + (1-1)(-3) = 13 + 0 = 13$
• For n=2: $a_2 = 13 + (2-1)(-3) = 13 + (1)(-3) = 13 - 3 = 10$
• For n=3: $a_3 = 13 + (3-1)(-3) = 13 + (2)(-3) = 13 - 6 = 7$
• For n=4: $a_4 = 13 + (4-1)(-3) = 13 + (3)(-3) = 13 - 9 = 4$
• For n=5: $a_5 = 13 + (5-1)(-3) = 13 + (4)(-3) = 13 - 12 = 1$

This confirms that the series follows the arithmetic progression rule with a common difference of -3, and the next term is indeed 1.