Understanding the 10th Term in the Sequence 3, 7, 11, 15

Have you ever come across a sequence of numbers and wondered what the 10th term would be? In this article, we'll delve into the sequence 3, 7, 11, 15 and explore how to determine its 10th term accurately. Whether you're a student or a professional looking to improve your understanding of number sequences, this guide will equip you with the knowledge and tools to solve such problems.

Identifying the Type of Sequence

The given sequence is an example of an arithmetic sequence (AP), a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. In the sequence 3, 7, 11, 15, the first term a is 3, and the common difference d is calculated as:

$$d 7 - 3 4$$

General Formula for the nth Term of an Arithmetic Sequence

The general formula for the nth term of an arithmetic sequence is given by:

$$t_n a (n - 1)d$$

Using this formula, we can easily find the 10th term (t_10) of our sequence.

Calculating the 10th Term

Substituting a 3 and d 4 into the formula, we get:

$$t_{10} 3 (10 - 1) times 4$$

Performing the calculations:

$$t_{10} 3 9 times 4 3 36 39$$

Therefore, the 10th term of the sequence 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, ... is 39.

Alternative Sequence: 1, 3, 6, 10, 15, 21, 28, 36, 45, ...

Distinct from the arithmetic sequence, the sequence 1, 3, 6, 10, 15, 21, 28, 36, 45, ... actually represents triangular numbers. Triangular numbers are a sequence where each number represents a triangle with dots. The formula to find the nth triangular number is:

$$T_n frac{n(n 1)}{2}$$

Finding the 10th Triangular Number

To find the 10th triangular number, substitute n 10 into the formula:

$$T_{10} frac{10(10 1)}{2} frac{10 times 11}{2} frac{110}{2} 55$$

Thus, the 10th triangular number is 55.

Conclusion

By understanding the concepts of arithmetic sequences and triangular numbers, you can now easily solve for the 10th term in both types of sequences. Whether you're working with arithmetic progressions or triangular numbers, the key is to identify the pattern and apply the correct formula.

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