Sequences & Series HPC Review

Front

What is an arithmetic sequence?

Back

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.

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How do you find the nth term of an arithmetic sequence?

Back

The nth term can be found using the formula: a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference.

Front

What is the common difference in an arithmetic sequence?

Back

The common difference is the constant amount that each term increases or decreases by in an arithmetic sequence.

Front

Given a_1 = 5 and d = 3, what is a_4?

Back

a_4 = 5 + (4-1)3 = 5 + 9 = 14.

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What is a convergent sequence?

Back

A convergent sequence is a sequence of numbers that approaches a specific value as the number of terms increases.

Front

What is a divergent sequence?

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A divergent sequence is a sequence that does not approach a specific value; it may increase or decrease without bound.

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How do you determine if a sequence converges or diverges?

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To determine convergence or divergence, analyze the behavior of the sequence as n approaches infinity.

Front

What is the explicit formula for the sequence 2, 8, 14, ...?

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The explicit formula is a_n = 6n - 4.

Front

If the 6th term of an arithmetic sequence is -38 and the 14th term is -110, what is the explicit rule?

Back

The explicit rule is a_n = 7 - 9(n - 1).

Front

What is the formula for the sum of the first n terms of an arithmetic sequence?

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The sum S_n of the first n terms is given by S_n = n/2 * (a_1 + a_n).