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There are four numbers of which the first three are in GP whose common ratio is 1/2 and the last three are in AP .If the last number is 2 less than the first find the four numbers.?
Verified Answer
There are four numbers of which the first three are in GP whose commo...
Let the 1st 3 terms be a-d,a,a+d
Ac. to Q. a-d+ a+d=2
Therefore, a=1
Now, let the last 3 terms be a,ar,ar2
Ac. to Q. a+ar2= 26
a(1+r2)=26
r2 =25
Therefore, r= +5 and -5
Therefore, the no.'s are -3,1,5,25 or 7,1,-5,25
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There are four numbers of which the first three are in GP whose commo...
Given Information:
- The first three numbers are in geometric progression (GP) with a common ratio of 1/2.
- The last three numbers are in arithmetic progression (AP).
- The last number is 2 less than the first number.
Let's solve this problem step by step:
Step 1: Understanding Geometric Progression (GP)
- In a geometric progression, each term is obtained by multiplying the previous term by a constant called the common ratio.
- The general form of a geometric progression is: a, ar, ar^2, ar^3, ...
where 'a' is the first term and 'r' is the common ratio.
Step 2: Finding the First Three Numbers
- Let the first number be 'a'.
- The second number would be 'ar' (since it is in GP).
- The third number would be 'ar^2' (since it is in GP).
Step 3: Understanding Arithmetic Progression (AP)
- In an arithmetic progression, each term is obtained by adding a constant called the common difference to the previous term.
- The general form of an arithmetic progression is: a, a+d, a+2d, a+3d, ...
where 'a' is the first term and 'd' is the common difference.
Step 4: Finding the Common Difference
- Since the last three numbers are in AP, the common difference would be 'd'.
- The last number would be 'a+2d' (since it is 2 less than the first number).
Step 5: Equating the Last Numbers
- Since the last number is 2 less than the first number, we can equate the expressions for the last number obtained in Steps 3 and 4.
(a+2d) = (a-2)
Step 6: Solving for 'a' and 'd'
- Solving the equation from Step 5, we get: a = 2d-2
Step 7: Finding the First Three Numbers (Continued)
- Substituting the value of 'a' from Step 6, we get: a = 2(2d)-2
Simplifying, we get: a = 4d-2
- The first three numbers are therefore: (4d-2), (2d-2), (2d-2)(1/2)
Step 8: Finding the Fourth Number
- The fourth number is obtained by adding the common difference to the third number, which is (2d-2)(1/2).
- The fourth number would be: [(2d-2)(1/2)] + d
Step 9: Equating the Fourth Number with the First Number
- Since the fourth number is equal to the first number, we can equate the expressions obtained in Steps 7 and 8.
(4d-2) = [(2d-2)(1/2)] + d
Step 10: Solving for 'd'
- Solving the equation from Step 9, we get: d = 6
Step 11: Finding the First Four Numbers
- The first three numbers are in geometric progression (GP) with a common ratio of 1/2.
- The last three numbers are in arithmetic progression (AP).
- The last number is 2 less than the first number.
Let's solve this problem step by step:
Step 1: Understanding Geometric Progression (GP)
- In a geometric progression, each term is obtained by multiplying the previous term by a constant called the common ratio.
- The general form of a geometric progression is: a, ar, ar^2, ar^3, ...
where 'a' is the first term and 'r' is the common ratio.
Step 2: Finding the First Three Numbers
- Let the first number be 'a'.
- The second number would be 'ar' (since it is in GP).
- The third number would be 'ar^2' (since it is in GP).
Step 3: Understanding Arithmetic Progression (AP)
- In an arithmetic progression, each term is obtained by adding a constant called the common difference to the previous term.
- The general form of an arithmetic progression is: a, a+d, a+2d, a+3d, ...
where 'a' is the first term and 'd' is the common difference.
Step 4: Finding the Common Difference
- Since the last three numbers are in AP, the common difference would be 'd'.
- The last number would be 'a+2d' (since it is 2 less than the first number).
Step 5: Equating the Last Numbers
- Since the last number is 2 less than the first number, we can equate the expressions for the last number obtained in Steps 3 and 4.
(a+2d) = (a-2)
Step 6: Solving for 'a' and 'd'
- Solving the equation from Step 5, we get: a = 2d-2
Step 7: Finding the First Three Numbers (Continued)
- Substituting the value of 'a' from Step 6, we get: a = 2(2d)-2
Simplifying, we get: a = 4d-2
- The first three numbers are therefore: (4d-2), (2d-2), (2d-2)(1/2)
Step 8: Finding the Fourth Number
- The fourth number is obtained by adding the common difference to the third number, which is (2d-2)(1/2).
- The fourth number would be: [(2d-2)(1/2)] + d
Step 9: Equating the Fourth Number with the First Number
- Since the fourth number is equal to the first number, we can equate the expressions obtained in Steps 7 and 8.
(4d-2) = [(2d-2)(1/2)] + d
Step 10: Solving for 'd'
- Solving the equation from Step 9, we get: d = 6
Step 11: Finding the First Four Numbers
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There are four numbers of which the first three are in GP whose common ratio is 1/2 and the last three are in AP .If the last number is 2 less than the first find the four numbers.?
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There are four numbers of which the first three are in GP whose common ratio is 1/2 and the last three are in AP .If the last number is 2 less than the first find the four numbers.? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about There are four numbers of which the first three are in GP whose common ratio is 1/2 and the last three are in AP .If the last number is 2 less than the first find the four numbers.? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for There are four numbers of which the first three are in GP whose common ratio is 1/2 and the last three are in AP .If the last number is 2 less than the first find the four numbers.?.
There are four numbers of which the first three are in GP whose common ratio is 1/2 and the last three are in AP .If the last number is 2 less than the first find the four numbers.? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about There are four numbers of which the first three are in GP whose common ratio is 1/2 and the last three are in AP .If the last number is 2 less than the first find the four numbers.? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for There are four numbers of which the first three are in GP whose common ratio is 1/2 and the last three are in AP .If the last number is 2 less than the first find the four numbers.?.
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