In Exercises 9–16, use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r.
Find a40 when a1 = 1000, r = - 1/2
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Identify the formula for the nth term of a geometric sequence: a_n = a_1 \cdot r^{n-1}.
Substitute the given values into the formula: a_1 = 1000, r = -\frac{1}{2}, and n = 40.
The formula becomes: a_{40} = 1000 \cdot \left(-\frac{1}{2}\right)^{39}.
Calculate the power: \left(-\frac{1}{2}\right)^{39}.
Multiply the result by 1000 to find a_{40}.
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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.
The general term formula for a geometric sequence allows us to calculate any term in the sequence based on its position. It is given by a_n = a_1 * r^(n-1). This formula is essential for finding specific terms, such as the 40th term in the sequence, by substituting the values of a_1, r, and n.
The common ratio in a geometric sequence is the factor by which we multiply each term to get the next term. It is denoted by r and can be positive or negative. In the given problem, the common ratio is -1/2, indicating that each term will alternate in sign and decrease in magnitude by half, which is crucial for accurately calculating the specified term.