Question

Write a recursive rule for the sequence. $$ 4,-12,36,-108, \ldots $$

Short answer

The recursive rule for the sequence is a_1 = 4, a_n = -3a_{n-1} for n > 1.

Step-by-step solution

Recognize the pattern in the sequence

It is observed in the given sequence 4, -12, 36, -108, that each subsequent number is 3 times the previous number and also alternates in sign.

Formulate recursive rule

Based on the pattern identified, the recursive rule for the given sequence can be formulated as following: a_1 = 4, a_n = -3a_{n-1} for n > 1 where 'a_n' represents the nth term of the sequence.

Verify the Rule

We can now apply this rule to the sequence and check it. a_1 = 4, for n=2, a_2=-3a_1=-3*4=-12, for n=3, a_3=-3a_2=-3*(-12)=36, and for n=4, a_4=-3a_3=-3*36=-108. This confirms that the derived rule is correct.

Key concepts

Recursive Formula

Recursive formulas are an important tool in mathematics, allowing us to define sequences by expressing each term based on the preceding one. This way of defining sequences is different from an explicit formula, which directly states the term based on its position in the sequence.

• A recursive formula consists of two parts: a starting value, and an equation to calculate subsequent terms.
• For the given sequence, the recursive formula starts with the initial term: \(a_1 = 4\). This means the first term is simply 4.
• Then, the formula provides a way to find any term, \(a_n\), using the previous term \(a_{n-1}\). For this sequence: \(a_n = -3a_{n-1}\) for \(n > 1\)

This approach is useful for sequences that follow a specific rule or pattern, as it simplifies the calculation of any term by just following the established procedure without needing to derive it independently each time.

Geometric Sequence

A geometric sequence is characterized by a constant ratio between successive terms. This constant ratio determines how each term relates to the previous one.

• For the sequence 4, -12, 36, -108, the ratio is -3.
• Notice that multiplying each term by -3 gives the next term in the sequence. For example, \(4 \times -3 = -12\) and \(-12 \times -3 = 36\).

In this sequence:

• The formula for finding any term is \(a_n = r\cdot a_{n-1}\), where \(r\) is the common ratio.
• This sequence exhibits exponential growth or decay, depending on the sign and value of \(r\).

It's important to note that geometric sequences can quickly grow in magnitude, demonstrating the power of exponential relationships.

Alternating Signs

Alternating signs in sequences mean that consecutive terms change their signs as they progress. This pattern can indicate multiplication by a negative number.

• In our sequence example, each term is produced by multiplying the previous term by -3, which automatically introduces a change in sign every time.
• For instance, from 4 to -12 is a positive to negative shift, then from -12 to 36 switches back to positive.

Understanding how alternating signs work can be very important, especially when calculating the terms from recursive relations.

• This means that if a sequence has a distinct pattern of alternating signs, it can often be attributed to operations involving negative multiplication.
• Recognizing alternating signs helps identify geometric sequences with negative ratios more easily.

The presence of alternating signs is a crucial characteristic to observe when analyzing any sequence, as it clues us into the operations governing the sequence.