10. Arithmetic and Geometric Sequences - Explicit Formula

Front

What is an arithmetic sequence?

Back

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

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How do you find the explicit formula for an arithmetic sequence?

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The explicit formula for an arithmetic sequence can be expressed as: \( a_n = a_1 + (n - 1) \cdot d \), where \( a_1 \) is the first term and \( d \) is the common difference.

Front

What is the common difference in the sequence 2, 8, 14, 20, ...?

Back

The common difference is 6, calculated as 8 - 2 or 14 - 8.

Front

Write the explicit formula for the arithmetic sequence 2, 8, 14, 20, ...

Back

The explicit formula is \( a_n = 2 + 6(n - 1) \).

Front

What is a geometric sequence?

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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.

Front

How do you find the explicit formula for a geometric sequence?

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The explicit formula for a geometric sequence can be expressed as: \( g_n = g_1 \cdot r^{(n - 1)} \), where \( g_1 \) is the first term and \( r \) is the common ratio.

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What is the common ratio in the geometric sequence 10, 20, 40, 80, ...?

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The common ratio is 2, calculated as 20 / 10 or 40 / 20.

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Write the explicit formula for the geometric sequence 10, 20, 40, 80, ...

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The explicit formula is \( g_n = 10(2)^{(n - 1)} \).

Front

What is the first term of the arithmetic sequence 15, 22, 29, 36, 43, ...?

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The first term is 15.

Front

What is the common difference in the sequence 15, 22, 29, 36, 43, ...?

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The common difference is 7, calculated as 22 - 15.