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a, b, c, d is an A.P. with integer terms and d = a2 + b2 + c2The value of a + 2b + 3c + 4d is
- a)0
- b)10
- c)100
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
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a, b, c, d is an A.P. with integer terms and d = a2 + b2 + c2The value...
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a, b, c, d is an A.P. with integer terms and d = a2 + b2 + c2The value...
Explanation:
Given
- a, b, c, d form an arithmetic progression with integer terms
- d = a^2 + b^2 + c^2
To find
Value of a + 2b + 3c + 4d
Calculations
- Let the common difference of the arithmetic progression be 'r'
- So, b = a + r and c = a + 2r
- Substituting these values in d = a^2 + b^2 + c^2, we get:
d = a^2 + (a + r)^2 + (a + 2r)^2
d = 3a^2 + 5ar + 5r^2
- Now, we need to find the value of a + 2b + 3c + 4d
= a + 2(a + r) + 3(a + 2r) + 4(3a^2 + 5ar + 5r^2)
= a + 2a + 2r + 3a + 6r + 12a + 10r + 15r^2
= 16a + 18r + 15r^2
- Since a, b, c, d are integers, the value of r must also be an integer
- To minimize the value of a + 2b + 3c + 4d, we can take a = 0, r = 1
- Substituting these values, we get:
= 16(0) + 18(1) + 15(1)^2
= 0 + 18 + 15
= 33
Therefore, the correct answer is option B) 10.
Given
- a, b, c, d form an arithmetic progression with integer terms
- d = a^2 + b^2 + c^2
To find
Value of a + 2b + 3c + 4d
Calculations
- Let the common difference of the arithmetic progression be 'r'
- So, b = a + r and c = a + 2r
- Substituting these values in d = a^2 + b^2 + c^2, we get:
d = a^2 + (a + r)^2 + (a + 2r)^2
d = 3a^2 + 5ar + 5r^2
- Now, we need to find the value of a + 2b + 3c + 4d
= a + 2(a + r) + 3(a + 2r) + 4(3a^2 + 5ar + 5r^2)
= a + 2a + 2r + 3a + 6r + 12a + 10r + 15r^2
= 16a + 18r + 15r^2
- Since a, b, c, d are integers, the value of r must also be an integer
- To minimize the value of a + 2b + 3c + 4d, we can take a = 0, r = 1
- Substituting these values, we get:
= 16(0) + 18(1) + 15(1)^2
= 0 + 18 + 15
= 33
Therefore, the correct answer is option B) 10.
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