Answer :
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.
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.
