Problem 52 - Algebra 2

Question

Use each recursive formula to write an explicit formula for the sequence. $$ a_{1}=10, a_{n}=2 a_{n-1} $$

Step-by-Step Solution

Answer

The explicit formula for the sequence is \(a_n = 10 * 2^{(n-1)}\).

Step 1: Identify the initial term

The initial term \(a_1\) is given as 10. This is the first term of the sequence.

Step 2: Identify the recurrence relation

The recurrence relation is \(a_{n}=2a_{n-1}\). This means the consequent term in the sequence is twice the previous term.

Step 3: Write down the first few terms

Using the initial term and recurrence relation, the first few terms of the sequence are: \(a_1 = 10, a_2 = 2a_{1} = 20, a_3 = 2a_{2} = 40, a_4 = 2a_{3} = 80\), and so on.

Step 4: Observe the pattern

The sequence is doubling each term, and thus it is geometric with a common ratio of 2.

Step 5: Write the explicit formula

Because the sequence is geometric, we can use the formula for the nth term of a geometric sequence: \(a_n = a_1 * r^{(n-1)}\), where \(r\) is the common ratio. So for our sequence, the explicit formula is \(a_n = 10 * 2^{(n-1)}\).

Key Concepts

recursive formula

A **recursive formula** helps us find each term in a sequence using preceding terms. It acts like a chain, where each link depends on the previous one. In the given exercise, the recursive formula is \( a_n = 2a_{n-1} \) with the initial term \( a_1 = 10 \). Here, every term is twice the previous term.
Recursive formulas follow a set pattern to express sequences:

• Start with an initial value: This is often given separately as part of the formula, like \( a_1 = 10 \) in the exercise.
• Define the relation: The formula \( a_n = 2a_{n-1} \) tells us how each term relates to the last.

This method is powerful for understanding how sequences evolve step-by-step. However, it requires previous terms to find the next one.

geometric sequence

A **geometric sequence** is a sequence where each term after the first is found by multiplying the previous one by a constant called the common ratio. In the exercise, the sequence starts with 10 and follows the formula \( a_n = 2a_{n-1} \). A few terms would be \( 10, 20, 40, 80 \), and so on. This pattern of multiplying by the same number shows we are dealing with a geometric sequence.
Features of geometric sequences include:

• The process is based on multiplication, such as multiplying by the common ratio.
• The growth is either exponential or decreasing, depending on whether the common ratio is greater or less than 1.

By understanding geometric sequences, one can easily predict future terms or derive an overall formula like the explicit formula given in the exercise.

common ratio

The **common ratio** is a key feature that defines a geometric sequence; it is the factor by which every term in the sequence is multiplied to obtain the next one. In our problem, the common ratio is \( 2 \), as seen from the recursive formulation \( a_n = 2a_{n-1} \). This means each term is double the previous term, leading to rapid exponential growth.
The common ratio has important characteristics:

• If greater than 1, the sequence grows exponentially.
• If between 0 and 1, the sequence shrinks or converges to zero.
• If negative, terms alternate their signs.

Knowing the common ratio helps summarize the entire sequence and is crucial in converting recursive formulas to explicit ones.

recurrence relation

A **recurrence relation** describes how terms in a sequence relate to previous terms. In essence, it's a formula that defines each term using its predecessors. For the given exercise, the recurrence relation is \( a_{n} = 2a_{n-1} \), which is easy to follow once you know the first term, \( a_1 = 10 \).
Common features of recurrence relations include:

• They often accompany an initial condition or a set of starting terms.
• They serve as the backbone for recursive sequences, indicating step-wise progress.
• Converting them into explicit formulas often leads to easier calculation of terms.

Recurrence relations exhibit the beauty and complexity of sequences, by showing how small changes can build a whole sequence.