Textbook Question
Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30.Find the sum of the first 14 terms of the geometric sequence: - 3/2, 3, - 6, 12, ...
9. Sequences, Series, & Induction
Geometric Sequences
2:56 minutesProblem 32
Textbook Question
In Exercises 31–34, write the first five terms of each geometric sequence. a1 = 1/2, r = 1/2
Verified step by step guidance1
Step 1: Understand the problem. A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant called the common ratio (r). Here, the first term (a₁) is 1/2, and the common ratio (r) is also 1/2.
Step 2: Recall the formula for the nth term of a geometric sequence: aₙ = a₁ * r^(n-1). This formula allows us to calculate any term in the sequence by substituting the values of a₁, r, and n.
Step 3: Calculate the second term (a₂). Substitute n = 2 into the formula: a₂ = a₁ * r^(2-1). This simplifies to a₂ = (1/2) * (1/2).
Step 4: Calculate the third term (a₃). Substitute n = 3 into the formula: a₃ = a₁ * r^(3-1). This simplifies to a₃ = (1/2) * (1/2)^2.
Step 5: Continue calculating the fourth term (a₄) and fifth term (a₅) using the same formula: a₄ = a₁ * r^(4-1) and a₅ = a₁ * r^(5-1). Substitute the values of a₁ and r into each expression to find the terms.
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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 is characterized by its exponential growth or decay, depending on whether the common ratio is greater than or less than one.
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First Term (a1)
The first term of a geometric sequence, denoted as a1, is the initial value from which the sequence begins. In this case, a1 = 1/2 indicates that the first term of the sequence is one-half, which serves as the foundation for generating subsequent terms through multiplication by the common ratio.
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Common Ratio (r)
The common ratio, denoted as r, is the factor by which each term in a geometric sequence is multiplied to obtain the next term. In this problem, r = 1/2 means that each term will be half of the previous term, leading to a sequence that decreases progressively.
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