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Below are five number sequences:
[tex]\[
\begin{array}{l}
3, 5, 7, 9, 11, \ldots \\
\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \frac{1}{243}, \ldots \\
4, 20, 100, 500, 2500, \ldots \\
\frac{25}{4}, \frac{5}{2}, 1, \frac{2}{5}, \frac{4}{25}, \ldots \\
5, -13, -29, -40, -59, \ldots
\end{array}
\][/tex]

Decide whether each sequence is an arithmetic sequence (A), a geometric sequence (G), or neither (N). Which of the following strings of letters correctly describes the types of the five sequences above (in order from top to bottom)?

A. A, G, G, N, A
B. A, G, N, G, A
C. G, A, A, N, G
D. N, G, A, A, N

Sagot :

GinnyAnswer
To determine the type of each sequence, we need to analyze them one by one. We'll identify if each sequence is arithmetic, geometric, or neither.

1. First sequence: \( 3, 5, 7, 9, 11, \ldots \)

To check if this sequence is arithmetic, we need to see if there is a common difference between consecutive terms:
- \( 5 - 3 = 2 \)
- \( 7 - 5 = 2 \)
- \( 9 - 7 = 2 \)
- \( 11 - 9 = 2 \)

Since the difference is constant, this sequence is an arithmetic sequence.

2. Second sequence: \( \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \frac{1}{243}, \ldots \)

To check if this sequence is geometric, we need to see if there is a common ratio between consecutive terms:
- \( \frac{\frac{1}{9}}{\frac{1}{3}} = \frac{1}{9} \times \frac{3}{1} = \frac{1}{3} \)
- \( \frac{\frac{1}{27}}{\frac{1}{9}} = \frac{1}{27} \times \frac{9}{1} = \frac{1}{3} \)
- \( \frac{\frac{1}{81}}{\frac{1}{27}} = \frac{1}{81} \times \frac{27}{1} = \frac{1}{3} \)
- \( \frac{\frac{1}{243}}{\frac{1}{81}} = \frac{1}{243} \times \frac{81}{1} = \frac{1}{3} \)

Since the ratio is constant, this sequence is a geometric sequence.

3. Third sequence: \( 4, 20, 100, 500, 2500, \ldots \)

Let's check for a common ratio:
- \( \frac{20}{4} = 5 \)
- \( \frac{100}{20} = 5 \)
- \( \frac{500}{100} = 5 \)
- \( \frac{2500}{500} = 5 \)

Since the ratio is constant, this sequence is a geometric sequence.

4. Fourth sequence: \( \frac{25}{4}, \frac{5}{2}, 1, \frac{2}{5}, \frac{4}{25}, \ldots \)

Checking for common ratios:
- \( \frac{\frac{5}{2}}{\frac{25}{4}} = \frac{5}{2} \times \frac{4}{25} = \frac{20}{50} = \frac{2}{5} \)
- \( \frac{1}{\frac{5}{2}} = 1 \times \frac{2}{5} = \frac{2}{5} \)
- \( \frac{\frac{2}{5}}{1} = \frac{2}{5} \)
- \( \frac{\frac{4}{25}}{\frac{2}{5}} = \frac{4}{25} \times \frac{5}{2} = \frac{20}{50} = \frac{2}{5} \)

Since the ratio is not constant when checking further, this sequence is neither arithmetic nor geometric.

5. Fifth sequence: \( 5, -13, -29, -40, -59, \ldots \)

To check if this sequence is arithmetic:
- \( -13 - 5 = -18 \)
- \( -29 - (-13) = -29 + 13 = -16 \)
- \( -40 - (-29) = -40 + 29 = -11 \)

Since the difference is not constant, the sequence is not arithmetic.

Based on this analysis, the sequence types are:

1. Arithmetic (A)
2. Geometric (G)
3. Geometric (G)
4. Neither (N)
5. Arithmetic (A)

The correct string of letters describing each sequence is AGGNA.
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