Problem 52 - Algebra 2

Question

Write an explicit and a recursive formula for each sequence. \(2,4,6,8,10, \dots\)

Step-by-Step Solution

Answer

The explicit formula for the sequence is \(a_n = 2n\) and the recursive formula for the sequence is \(a_n = a_{n-1} + 2\), where \(a_1 = 2\).

Step 1: Identify the first term and the common difference

The first term \(a_1\) in the sequence is 2 and the common difference \(d\), which is the difference between successive terms, is 2 (4-2).

Step 2: Write the explicit formula

Substitute \(a_1 = 2\) and \(d = 2\) into the explicit formula \(a_n = a_1 + (n-1)d\) to get \(a_n = 2 + (n-1)2 = 2n\). Therefore, the explicit formula for the sequence is \(a_n = 2n\).

Step 3: Write the recursive formula

Substitute \(a_1 = 2\) and \(d = 2\) into the recursive formula \(a_n = a_{n-1} + d\) to get \(a_n = a_{n-1} + 2\). Therefore, the recursive formula for the sequence is \(a_n = a_{n-1} + 2\) where \(a_1 = 2\).

Key Concepts

Explicit Formula

An explicit formula is a powerful way to describe a sequence. It allows you to compute any term in the sequence by substitution without knowing any previous terms. For the sequence given, where the numbers are 2, 4, 6, 8, 10, and so on, the explicit formula helps in providing a direct computation for any term number you might be interested in.

To form an explicit formula, you typically need to know the first term of the sequence and the common difference when dealing with arithmetic sequences. For our sequence, we have:

• First term (\(a_1\)): 2
• Common difference (\(d\)): 2

Plug these values into the standard form of an arithmetic sequence explicit formula:\[ a_n = a_1 + (n-1)d \]By substituting the values we have:\[ a_n = 2 + (n-1) imes 2 \] Simplifying gives:\[ a_n = 2n \]Thus, for any given term number \(n\), the value of that term can be quickly found by multiplying \(n\) by 2.

Recursive Formula

If you enjoy step-by-step processes, the recursive formula might be your go-to. Recursive formulas express each term in a sequence based on the preceding term. This is particularly useful if you need to calculate terms in order and you're given the first term.

For our sequence, we have:

• First term (\(a_1\)): 2
• Common difference (\(d\)): 2

With this knowledge, we can form a recursive formula. The basic idea is:\[ a_n = a_{n-1} + d \] Where \(a_{n-1}\) is the term before \(a_n\). For our sequence:\[ a_n = a_{n-1} + 2 \]Starting with \(a_1 = 2\), you can find any subsequent term by adding 2 to the previous term. If \(a_2 = 4\), then \(a_3\) is \(4 + 2\), which equals 6, and so on.
This method is seen as more of a process than a direct calculation and shines when calculating terms sequentially.

Arithmetic Sequence

Sequences are a big part of mathematics, and arithmetic sequences are a special type of sequence. An arithmetic sequence is one in which every term is generated by adding a constant amount, known as the common difference, to the previous term. They are easy to recognize by their consistent rates of change.

For an arithmetic sequence, the pattern is consistent. You're adding the same number at each step to find the next term. For our sequence, the rule is:

• Start with 2
• Add 2 to get the next term: 2, 4, 6, 8, 10...

The beauty of an arithmetic sequence lies in its simplicity and predictability. Once you know the first term and the common difference, the whole sequence unrolls effortlessly. It's like a predictable pattern that forms a straight line if you were to plot the terms on a graph, hence it's also known as linear sequences.
Understanding these types of sequences can greatly support your mathematical intuition, especially when progressing to more complex topics.