Unit 1: Sequences Review

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What is a recursive sequence?

Back

A recursive sequence is a sequence of numbers where each term is defined as a function of the preceding terms. For example, in the sequence defined by \( a_1 = 39 \) and \( a_n = a_{n-1} + 4 \), each term is generated by adding 4 to the previous term.

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How do you find the nth term of a recursive sequence?

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To find the nth term of a recursive sequence, you start with the initial term and apply the recursive formula repeatedly until you reach the desired term. For example, for \( a_n = a_{n-1} + 4 \) with \( a_1 = 39 \), the 52nd term is \( a_{52} = 39 + 4 \times (52-1) = 243 \).

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What is the formula for an arithmetic sequence?

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The formula for the nth term of an arithmetic sequence is given by \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

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What is the common difference in an arithmetic sequence?

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The common difference in an arithmetic sequence is the constant amount that each term increases or decreases by. For example, in the sequence 39, 43, 47, the common difference is 4.

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How do you determine the 10th term of a sequence given a pattern?

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To determine the 10th term of a sequence given a pattern, identify the rule or formula that generates the sequence and apply it to find the 10th term. For example, if the pattern leads to 1536 for the 10th term, the rule must be derived from the previous terms.

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What is the cost function for a job that charges a flat fee plus a variable rate?

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The cost function can be expressed as \( C(n) = C_0 + r \cdot n \), where \( C_0 \) is the initial fee, \( r \) is the rate per hour, and \( n \) is the number of hours worked. For example, \( C(n) = 250 + 20n \) represents a flat fee of $250 plus $20 per hour.

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How do you calculate the total cost for multiple hours of work?

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To calculate the total cost for multiple hours of work, use the cost function and substitute the number of hours into the equation. For example, for 4 hours, \( C(4) = 250 + 20 \times 4 = 330 \).

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What is the first term of the recursive sequence defined by \( a_1 = 39 \)?

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The first term of the recursive sequence defined by \( a_1 = 39 \) is simply 39.

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How do you find the 22nd term of a recursive sequence?

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To find the 22nd term of a recursive sequence, apply the recursive formula starting from the first term. For example, \( a_{22} = 39 + 4 \times (22-1) = 123 \).

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What is the significance of the initial term in a recursive sequence?

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The initial term in a recursive sequence serves as the starting point for generating all subsequent terms. It is crucial for determining the values of all other terms in the sequence.