Question

In Problems 22-25, find the first four terms of the sequence and a formula for the general term. $$ a_{n}=2 a_{n-1} ; a_{1}=1 $$

Short answer

Answer: The first four terms of the sequence are 1, 2, 4, 8. The explicit formula for the general term of the sequence is $$a_n = 2^{n-1}$$.

Step-by-step solution

Find the first four terms of the sequence using the given recursive formula

To find the first four terms of the sequence, we will need to use the recursive formula $$a_n = 2 a_{n-1}$$ and the initial term $$a_1 = 1$$, calculating each term based on the previous term. - For the first term, we have: $$a_1 = 1$$ - Now, we find the second term using the formula with n = 2: $$a_2 = 2 a_{1} = 2(1) = 2$$ - Next, we find the third term with n = 3: $$a_3 = 2 a_{2} = 2(2) = 4$$ - Finally, we find the fourth term with n = 4: $$a_4 = 2 a_{3} = 2(4) = 8$$ So, the first four terms of the sequence are 1, 2, 4, 8.

Find the explicit formula for the general term

To find the explicit formula, we can recognize this sequence as a geometric series, where each term is multiplied by a constant factor to get the next term (in this case, the factor is 2). Therefore, the general formula for this sequence should be of the form: $$a_n = a_1 * r^{n-1}$$, where 'r' is the common ratio. Here, we can see that the common ratio 'r' is 2 (since each term is multiplied by 2 to get the next term) and the initial term $$a_1$$ is 1. Plugging these values into the general formula, we get: $$a_n = 1 * 2^{n-1}$$. Thus, the explicit formula for the general term of this sequence is $$a_n = 2^{n-1}$$.

Key concepts

Recursive Formula

A recursive formula defines each term of a sequence using the preceding term or terms. Essentially, it allows us to generate the subsequent terms based on an initial value known as the initial term. In our example, the recursive formula provided is:

• \(a_n = 2a_{n-1}\)

This means each term \(a_n\) is calculated by multiplying the previous term \(a_{n-1}\) by 2. The sequence starts with \(a_1 = 1\), thus making it the foundation for the following terms. Recursive formulas like this one help us understand the relationship between the sequence terms. However, to find later terms quickly, we often look for an explicit formula.

Explicit Formula

The explicit formula gives a direct way to calculate any term in a sequence without needing the previous term. It is particularly useful for finding terms that are far into the sequence. In a geometric sequence like ours, the explicit formula can be derived from recognizing the pattern of constant multiplication. Using the given sequence's first four terms—1, 2, 4, and 8—the explicit formula is:

• \(a_n = 2^{n-1}\)

Using this formula, you can determine the value of any term \(a_n\) simply by plugging in the term number \(n\). This is helpful when dealing with large term numbers where finding each intervening term recursively would be impractical.

Geometric Series

A geometric series is the sum of the terms of a geometric sequence. In a geometric sequence, each term is obtained by multiplying the previous one by a constant called the common ratio. For the sequence in our problem, the common ratio \(r\) is 2. The terms—1, 2, 4, and 8—form a geometric series, and the sum of these terms can be calculated using the formula for the sum of the first \(n\) terms of a geometric series:

• \(S_n = a_1 \frac{r^n - 1}{r - 1}\)

Here, \(a_1 = 1\) and \(r = 2\). We see geometric series utilized in various fields, from mathematics and physics to finance, as they help model exponential growth processes.