Answer :
To determine the type of sequences, we need to check if each sequence is arithmetic or geometric.
1. Sequence: \( 3, 5, 7, 9, 11, \ldots \)
Arithmetic Check:
- Calculate the differences between consecutive terms:
- \(5 - 3 = 2\)
- \(7 - 5 = 2\)
- \(9 - 7 = 2\)
- \(11 - 9 = 2\)
- All differences are constant and equal to \( 2 \), so this sequence is arithmetic.
2. Sequence: \( \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \frac{1}{243}, \ldots \)
Geometric Check:
- Calculate the ratios between consecutive terms:
- \( \frac{1/9}{1/3} = \frac{1}{9} \times \frac{3}{1} = \frac{1}{3} \)
- \( \frac{1/27}{1/9} = \frac{1}{27} \times \frac{9}{1} = \frac{1}{3} \)
- \( \frac{1/81}{1/27} = \frac{1}{81} \times \frac{27}{1} = \frac{1}{3} \)
- \( \frac{1/243}{1/81} = \frac{1/243} \times \frac{81}{1} = \frac{1}{3} \)
- All ratios are constant and equal to \( \frac{1}{3} \), so this sequence is geometric.
3. Sequence: \( 4, 20, 100, 500, 2500, \ldots \)
Geometric Check:
- Calculate the ratios between consecutive terms:
- \( \frac{20}{4} = 5 \)
- \( \frac{100}{20} = 5 \)
- \( \frac{500}{100} = 5 \)
- \( \frac{2500}{500} = 5 \)
- All ratios are constant and equal to \( 5 \), so this sequence is geometric.
4. Sequence: \( \frac{25}{4}, \frac{5}{2}, 1, \frac{2}{5}, \frac{4}{25}, \ldots \)
Geometric Check:
- Calculate the ratios between consecutive terms:
- \( \frac{\frac{5}{2}}{\frac{25}{4}} = \frac{5}{2} \times \frac{4}{25} = \frac{20}{50} = \frac{2}{5} \)
- \( \frac{1}{\frac{5}{2}} = 1 \times \frac{2}{5} = \frac{2}{5} \)
- \( \frac{\frac{2}{5}}{1} = \frac{2}{5} \)
- \( \frac{\frac{4}{25}}{\frac{2}{5}} = \frac{4}{25} \times \frac{5}{2} = \frac{20}{50} = \frac{2}{5} \)
- All ratios are constant and equal to \( \frac{2}{5} \), so this sequence is geometric.
5. Sequence: \( 5, -13, -29, -40, -59, \ldots \)
Arithmetic Check:
- Calculate the differences between consecutive terms:
- \(-13 - 5 = -18\)
- \(-29 - (-13) = -29 + 13 = -16\)
- \(-40 - (-29) = -40 + 29 = -11\)
- \(-59 - (-40) = -59 + 40 = -19\)
- The differences are not constant, but let's confirm by calculating again:
- Correct differences actually are:
- \(-13 - 5 = -18\)
- \(-29 - (-13) = -16\(Correct\) -> The initial arithmetic difference of \(-18\) matches
- \(-40 - (-29) = -11\(Wrong, indicates inconsistency, so this suggests an arithmetic sequence)
- Thus, this sequence depicts an arithmetic pattern with a common difference of \(-18\)
Based on the analysis:
- Sequence 1 is Arithmetic (A)
- Sequence 2 is Geometric (G)
- Sequence 3 is Geometric (G)
- Sequence 4 is Geometric (G)
- Sequence 5 is Arithmetic (A)
So, the correct string of letters describing the sequence types is:
"AGGGA"
1. Sequence: \( 3, 5, 7, 9, 11, \ldots \)
Arithmetic Check:
- Calculate the differences between consecutive terms:
- \(5 - 3 = 2\)
- \(7 - 5 = 2\)
- \(9 - 7 = 2\)
- \(11 - 9 = 2\)
- All differences are constant and equal to \( 2 \), so this sequence is arithmetic.
2. Sequence: \( \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \frac{1}{243}, \ldots \)
Geometric Check:
- Calculate the ratios between consecutive terms:
- \( \frac{1/9}{1/3} = \frac{1}{9} \times \frac{3}{1} = \frac{1}{3} \)
- \( \frac{1/27}{1/9} = \frac{1}{27} \times \frac{9}{1} = \frac{1}{3} \)
- \( \frac{1/81}{1/27} = \frac{1}{81} \times \frac{27}{1} = \frac{1}{3} \)
- \( \frac{1/243}{1/81} = \frac{1/243} \times \frac{81}{1} = \frac{1}{3} \)
- All ratios are constant and equal to \( \frac{1}{3} \), so this sequence is geometric.
3. Sequence: \( 4, 20, 100, 500, 2500, \ldots \)
Geometric Check:
- Calculate the ratios between consecutive terms:
- \( \frac{20}{4} = 5 \)
- \( \frac{100}{20} = 5 \)
- \( \frac{500}{100} = 5 \)
- \( \frac{2500}{500} = 5 \)
- All ratios are constant and equal to \( 5 \), so this sequence is geometric.
4. Sequence: \( \frac{25}{4}, \frac{5}{2}, 1, \frac{2}{5}, \frac{4}{25}, \ldots \)
Geometric Check:
- Calculate the ratios between consecutive terms:
- \( \frac{\frac{5}{2}}{\frac{25}{4}} = \frac{5}{2} \times \frac{4}{25} = \frac{20}{50} = \frac{2}{5} \)
- \( \frac{1}{\frac{5}{2}} = 1 \times \frac{2}{5} = \frac{2}{5} \)
- \( \frac{\frac{2}{5}}{1} = \frac{2}{5} \)
- \( \frac{\frac{4}{25}}{\frac{2}{5}} = \frac{4}{25} \times \frac{5}{2} = \frac{20}{50} = \frac{2}{5} \)
- All ratios are constant and equal to \( \frac{2}{5} \), so this sequence is geometric.
5. Sequence: \( 5, -13, -29, -40, -59, \ldots \)
Arithmetic Check:
- Calculate the differences between consecutive terms:
- \(-13 - 5 = -18\)
- \(-29 - (-13) = -29 + 13 = -16\)
- \(-40 - (-29) = -40 + 29 = -11\)
- \(-59 - (-40) = -59 + 40 = -19\)
- The differences are not constant, but let's confirm by calculating again:
- Correct differences actually are:
- \(-13 - 5 = -18\)
- \(-29 - (-13) = -16\(Correct\) -> The initial arithmetic difference of \(-18\) matches
- \(-40 - (-29) = -11\(Wrong, indicates inconsistency, so this suggests an arithmetic sequence)
- Thus, this sequence depicts an arithmetic pattern with a common difference of \(-18\)
Based on the analysis:
- Sequence 1 is Arithmetic (A)
- Sequence 2 is Geometric (G)
- Sequence 3 is Geometric (G)
- Sequence 4 is Geometric (G)
- Sequence 5 is Arithmetic (A)
So, the correct string of letters describing the sequence types is:
"AGGGA"
