what is the arithmetic sequence of 2, 4, 6, 8, 10, 12…
The correct answer and explanation is:
The given sequence is 2, 4, 6, 8, 10, 12, and so on. This is an arithmetic sequence because each term increases by a constant difference.
Correct Answer:
- The first term a1=2a_1 = 2
- The common difference d=4−2=2d = 4 – 2 = 2
- The general term (nth term) of the arithmetic sequence is given by the formula:
an=a1+(n−1)da_n = a_1 + (n – 1)d
Substituting the values: an=2+(n−1)×2=2+2n−2=2na_n = 2 + (n – 1) \times 2 = 2 + 2n – 2 = 2n
Thus, the nn-th term of this arithmetic sequence is: an=2na_n = 2n
Explanation:
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant is called the common difference. In this sequence, every term increases by 2 from the previous one. Starting from 2, the sequence goes 2, 4, 6, 8, 10, 12, etc.
The first term is the initial number of the sequence. Here, the first term a1a_1 is 2. To find the general term ana_n, which represents any term at position nn, the formula uses the first term and adds the common difference multiplied by one less than the term’s position because the first term is at n=1n = 1.
Calculating the second term: a2=2+(2−1)×2=2+2=4a_2 = 2 + (2-1) \times 2 = 2 + 2 = 4.
Calculating the third term: a3=2+(3−1)×2=2+4=6a_3 = 2 + (3-1) \times 2 = 2 + 4 = 6.
This confirms the pattern.
Since the general term simplifies to an=2na_n = 2n, the sequence consists of even numbers starting from 2 and increasing by 2 each time.
Arithmetic sequences are important in many areas such as mathematics, physics, finance, and computer science because they represent situations with steady, uniform growth or decline. Recognizing the pattern and using the general term formula allows predicting any term without listing all previous terms.
In summary, this sequence is an arithmetic sequence with first term 2 and common difference 2, and its general term is an=2na_n = 2n.