In Exercises 1–8, write the first five terms of each geometric sequence.
an = - 4a_(n-1), a1 = 10
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1
Identify the first term of the sequence, which is given as .
Recognize that the sequence is geometric, meaning each term is obtained by multiplying the previous term by a constant ratio.
The recursive formula for the sequence is . This indicates that each term is obtained by multiplying the previous term by .
Calculate the second term: .
Continue this process to find the next terms: , , and .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Geometric Sequence
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This type of sequence can be expressed in the form a_n = a_1 * r^(n-1), where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the term number.
A recursive formula defines each term of a sequence based on the preceding term(s). In the given question, the formula an = -4a_(n-1) indicates that each term is derived by multiplying the previous term by -4. This approach is essential for generating terms in sequences where the relationship between terms is defined recursively rather than explicitly.
The initial term of a sequence is the first term from which all subsequent terms are generated. In this case, a1 = 10 serves as the starting point for the geometric sequence. Understanding the initial term is crucial because it directly influences the values of all following terms in the sequence.