Discover the answers you need at Westonci.ca, a dynamic Q&A platform where knowledge is shared freely by a community of experts. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
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"
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
