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In how many ways can we select three natural numbers out of the first 10 natural numbers so that they are in a geometric progression with the common ratio greater than 1?
- a)2 ways
- b)3 ways
- c)4 ways
- d)5 ways
Correct answer is option 'B'. Can you explain this answer?
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In how many ways can we select three natural numbers out of the first ...
Make pair of three numbers. Only sequence with 2 and 3 as the common ratio is possible.
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In how many ways can we select three natural numbers out of the first ...
1,2,4 # 2,4,8# 1,3,9 are three ways to select with ratio of 2,2 and 3 in GP
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In how many ways can we select three natural numbers out of the first ...
To solve this problem, we need to find the number of ways we can select three natural numbers out of the first 10 natural numbers such that they are in a geometric progression with a common ratio greater than 1.
Let's consider the possible common ratios that can be formed using the first 10 natural numbers:
1) Common Ratio = 2:
To form a geometric progression with a common ratio of 2, we can select the numbers {1, 2, 4}, {1, 4, 8}, or {2, 4, 8}. So, we have 3 possible selections.
2) Common Ratio = 3:
To form a geometric progression with a common ratio of 3, we can select the numbers {1, 3, 9}. So, we have 1 possible selection.
3) Common Ratio = 4:
To form a geometric progression with a common ratio of 4, we can select the numbers {1, 4, 16}. However, 16 is not one of the first 10 natural numbers. So, there are no possible selections.
4) Common Ratio = 5:
To form a geometric progression with a common ratio of 5, we can select the numbers {1, 5, 25}. However, 25 is not one of the first 10 natural numbers. So, there are no possible selections.
5) Common Ratio = 6:
To form a geometric progression with a common ratio of 6, we can select the numbers {1, 6, 36}. However, 36 is not one of the first 10 natural numbers. So, there are no possible selections.
6) Common Ratio = 7:
To form a geometric progression with a common ratio of 7, we can select the numbers {1, 7, 49}. However, 49 is not one of the first 10 natural numbers. So, there are no possible selections.
7) Common Ratio = 8:
To form a geometric progression with a common ratio of 8, we can select the numbers {1, 8, 64}. However, 64 is not one of the first 10 natural numbers. So, there are no possible selections.
8) Common Ratio = 9:
To form a geometric progression with a common ratio of 9, we can select the numbers {1, 9, 81}. However, 81 is not one of the first 10 natural numbers. So, there are no possible selections.
9) Common Ratio = 10:
To form a geometric progression with a common ratio of 10, we can select the numbers {1, 10, 100}. However, 100 is not one of the first 10 natural numbers. So, there are no possible selections.
From the above analysis, we can see that there are a total of 1 + 3 = 4 possible selections.
Therefore, the correct answer is option C) 4 ways.
Let's consider the possible common ratios that can be formed using the first 10 natural numbers:
1) Common Ratio = 2:
To form a geometric progression with a common ratio of 2, we can select the numbers {1, 2, 4}, {1, 4, 8}, or {2, 4, 8}. So, we have 3 possible selections.
2) Common Ratio = 3:
To form a geometric progression with a common ratio of 3, we can select the numbers {1, 3, 9}. So, we have 1 possible selection.
3) Common Ratio = 4:
To form a geometric progression with a common ratio of 4, we can select the numbers {1, 4, 16}. However, 16 is not one of the first 10 natural numbers. So, there are no possible selections.
4) Common Ratio = 5:
To form a geometric progression with a common ratio of 5, we can select the numbers {1, 5, 25}. However, 25 is not one of the first 10 natural numbers. So, there are no possible selections.
5) Common Ratio = 6:
To form a geometric progression with a common ratio of 6, we can select the numbers {1, 6, 36}. However, 36 is not one of the first 10 natural numbers. So, there are no possible selections.
6) Common Ratio = 7:
To form a geometric progression with a common ratio of 7, we can select the numbers {1, 7, 49}. However, 49 is not one of the first 10 natural numbers. So, there are no possible selections.
7) Common Ratio = 8:
To form a geometric progression with a common ratio of 8, we can select the numbers {1, 8, 64}. However, 64 is not one of the first 10 natural numbers. So, there are no possible selections.
8) Common Ratio = 9:
To form a geometric progression with a common ratio of 9, we can select the numbers {1, 9, 81}. However, 81 is not one of the first 10 natural numbers. So, there are no possible selections.
9) Common Ratio = 10:
To form a geometric progression with a common ratio of 10, we can select the numbers {1, 10, 100}. However, 100 is not one of the first 10 natural numbers. So, there are no possible selections.
From the above analysis, we can see that there are a total of 1 + 3 = 4 possible selections.
Therefore, the correct answer is option C) 4 ways.
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In how many ways can we select three natural numbers out of the first 10 natural numbers so that they are in a geometric progression with the common ratio greater than 1?a)2 waysb)3 waysc)4 waysd)5 waysCorrect answer is option 'B'. Can you explain this answer?
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