Let {an} = - 5, 10, - 20, 40, ..., {bn} = 10, - 5, - 20, - 35, ..., {cn} = - 2, 1, - 1/2, 1/4 Find a10 + b10.

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 57

9장, 문제 57

Let {a_n} = - 5, 10, - 20, 40, ..., {b_n} = 10, - 5, - 20, - 35, ..., {c_n} = - 2, 1, - 1/2, 1/4 Find a₁₀ + b₁₀.

검증된 단계별 안내

Identify the type of each sequence \( \{a_n\} \), \( \{b_n\} \), and \( \{c_n\} \) by examining the pattern of their terms. For example, check if they are arithmetic (constant difference) or geometric (constant ratio) sequences.

For the sequence \( \{a_n\} = -5, 10, -20, 40, \ldots \), calculate the ratio between consecutive terms to determine if it is geometric. Use the formula for the \(n\)-th term of a geometric sequence: \[ a_n = a_1 \times r^{n-1} \] where \(a_1\) is the first term and \(r\) is the common ratio.

For the sequence \( \{b_n\} = 10, -5, -20, -35, \ldots \), check if the difference between consecutive terms is constant to see if it is arithmetic. Use the formula for the \(n\)-th term of an arithmetic sequence: \[ b_n = b_1 + (n-1)d \] where \(b_1\) is the first term and \(d\) is the common difference.

Calculate \( a_{10} \) using the formula found in step 2 and \( b_{10} \) using the formula found in step 3 by substituting \( n = 10 \).

Add the two results together to find \( a_{10} + b_{10} \). This sum is the final answer you are asked to find.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Sequences and Terms

A sequence is an ordered list of numbers following a specific pattern. Each number in the sequence is called a term, denoted by a subscript (e.g., aₙ). Understanding how to identify and work with terms is essential for finding specific values like a₁₀ or b₁₀.

Arithmetic and Geometric Sequences

Sequences often follow arithmetic (constant difference) or geometric (constant ratio) patterns. Recognizing the type helps determine the formula for the nth term, which is crucial for calculating terms far along the sequence, such as the 10th term.

Finding the nth Term Formula

To find a specific term like a₁₀, you need the nth term formula. For arithmetic sequences, use aₙ = a₁ + (n-1)d; for geometric sequences, use aₙ = a₁ * r^(n-1). Identifying the pattern and applying the correct formula allows calculation of any term.