Understanding Arithmetic Sequences: General Term of 2, 4, 6, 8, 10

Introduction to Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which each term, after the first, is obtained by adding a constant value, known as the common difference, to the preceding term. For instance, in the sequence 2, 4, 6, 8, 10, each number increases by 2.

Determining the General Term

To determine the general term of the arithmetic sequence, we can use the formula:

Mathematical Explanation and Formula

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

an a1 (n-1)d

where:

an is the nth term of the sequence. a1 is the first term of the sequence. d is the common difference between terms. n is the position of the term in the sequence.

For the sequence 2, 4, 6, 8, 10:

a1 2 d 2

Let's derive the general formula for this specific sequence:

2, 4, 6, 8, 10

The first term, a1, is 2. The common difference, d, is 2.

Substituting these values into the formula:

an 2 (n-1)2

Simplifying:

an 2 2n - 2

an 2n

Therefore, the general term of the sequence is an 2n.

Prove by Substitution

To verify the general term, we can substitute different values of n into the formula:

Example:

n 1: a1 2 * 1 2

n 2: a2 2 * 2 4

n 3: a3 2 * 3 6

n 4: a4 2 * 4 8

n 5: a5 2 * 5 10

These values match the terms in the sequence exactly, confirming the general term an 2n.

Pattern Recognition

Examining the difference between consecutive terms:

4 - 2 2 6 - 4 2 8 - 6 2 10 - 8 2

The difference is consistently 2, which is the common difference (d) used in the general term formula.

Another way to represent the sequence is to observe the pattern in the terms:

n 0: 2 * 0 0 n 1: 2 * 1 2 n 2: 2 * 2 4 n 3: 2 * 3 6 n 4: 2 * 4 8 n 5: 2 * 5 10

This confirms the general term formula an 2n.

Conclusion

The general term of the arithmetic sequence 2, 4, 6, 8, 10 is given by an 2n. This formula can be used to find any term in the sequence based on its position in the sequence.