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If a,b,c,d are in ap & a,c,d are in gp so prove that a^2-d^2=3 (b^2-ad)?
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If a,b,c,d are in ap & a,c,d are in gp so prove that a^2-d^2=3 (b^2-ad...
If a, b, c, and d are in arithmetic progression (AP), it means that the difference between any two consecutive terms is the same. Let's say that the common difference is denoted by 'd'.
In an arithmetic progression, the general formula for the nth term is given by:
an = a + (n-1)d
Using this formula, we can find the value of any term in the AP.
For example, the second term (b) can be expressed as:
b = a + (2-1)d
b = a + d
Similarly, the third term (c) can be expressed as:
c = a + (3-1)d
c = a + 2d
And the fourth term (d) can be expressed as:
d = a + (4-1)d
d = a + 3d
d = 4d
So, if a, b, c, and d are in arithmetic progression, we can express each term in terms of the first term (a) and the common difference (d) as follows:
b = a + d
c = a + 2d
d = a + 3d
Note: The values of a, b, c, and d are not specified in the question, so we cannot determine them without additional information.
In an arithmetic progression, the general formula for the nth term is given by:
an = a + (n-1)d
Using this formula, we can find the value of any term in the AP.
For example, the second term (b) can be expressed as:
b = a + (2-1)d
b = a + d
Similarly, the third term (c) can be expressed as:
c = a + (3-1)d
c = a + 2d
And the fourth term (d) can be expressed as:
d = a + (4-1)d
d = a + 3d
d = 4d
So, if a, b, c, and d are in arithmetic progression, we can express each term in terms of the first term (a) and the common difference (d) as follows:
b = a + d
c = a + 2d
d = a + 3d
Note: The values of a, b, c, and d are not specified in the question, so we cannot determine them without additional information.
Community Answer
If a,b,c,d are in ap & a,c,d are in gp so prove that a^2-d^2=3 (b^2-ad...
Since a,b,c,d are in AP (with common difference r),
a + d = a + (a + 3r) = 2a + 3r ---(1)
b + c = (a + r) + (a + 2r) = 2a + 3r
Also, a - d = -3r ---(2) and b - c = -r
Since a,c,d are in GP => c² = ad
(1) x (2) = a² - d² = -3r(2a + 3r) ---(3)
3(b² - ad) = 3(b² - c²) = (b + c)(b - c) = (2a + 3r)(-r) = -3r(2a + 3r) ---(4)
(3) = (4)
Hence proved
a + d = a + (a + 3r) = 2a + 3r ---(1)
b + c = (a + r) + (a + 2r) = 2a + 3r
Also, a - d = -3r ---(2) and b - c = -r
Since a,c,d are in GP => c² = ad
(1) x (2) = a² - d² = -3r(2a + 3r) ---(3)
3(b² - ad) = 3(b² - c²) = (b + c)(b - c) = (2a + 3r)(-r) = -3r(2a + 3r) ---(4)
(3) = (4)
Hence proved
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If a,b,c,d are in ap & a,c,d are in gp so prove that a^2-d^2=3 (b^2-ad)?
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If a,b,c,d are in ap & a,c,d are in gp so prove that a^2-d^2=3 (b^2-ad)? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about If a,b,c,d are in ap & a,c,d are in gp so prove that a^2-d^2=3 (b^2-ad)? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a,b,c,d are in ap & a,c,d are in gp so prove that a^2-d^2=3 (b^2-ad)?.
If a,b,c,d are in ap & a,c,d are in gp so prove that a^2-d^2=3 (b^2-ad)? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about If a,b,c,d are in ap & a,c,d are in gp so prove that a^2-d^2=3 (b^2-ad)? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a,b,c,d are in ap & a,c,d are in gp so prove that a^2-d^2=3 (b^2-ad)?.
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