Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
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"
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.