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The sum of 5 numbers in AP is 30 and the sum of their squares is 220. Which of the following is the third term?
- a)5
- b)6
- c)8
- d)9
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The sum of 5 numbers in AP is 30 and the sum of their squares is 220. ...
Since the sum of 5 numbers in AP is 30, their average would be 6. The average of 5 terms in AP is
also equal to the value of the 3rd term (logic of the middle term of an AP). Hence, the third term’s
value would be 6. Option (b) is correct.
also equal to the value of the 3rd term (logic of the middle term of an AP). Hence, the third term’s
value would be 6. Option (b) is correct.
Most Upvoted Answer
The sum of 5 numbers in AP is 30 and the sum of their squares is 220. ...
Solution:
Let the first term of the AP be a and the common difference be d.
Then the five numbers are: a, a+d, a+2d, a+3d, a+4d
Given that the sum of the five numbers is 30:
a + a+d + a+2d + a+3d + a+4d = 30
5a + 10d = 30
a + 2d = 6
Also given that the sum of their squares is 220:
a^2 + (a+d)^2 + (a+2d)^2 + (a+3d)^2 + (a+4d)^2 = 220
5a^2 + 30d^2 + 20ad = 220
a^2 + 6d^2 + 4ad = 44
Substituting a+2d=6, we get:
a = 6 - 2d
Substituting this value of a in the above equation, we get:
(6-2d)^2 + 6d^2 + 4(6-2d)d = 44
Simplifying this equation, we get:
4d^2 - 8d - 5 = 0
On solving this quadratic equation, we get:
d = (-(-8) ± √((-8)^2 - 4(4)(-5))) / (2(4))
d = (8 ± √116) / 8
d = (2 ± √29) / 2
Since the common difference of an AP cannot be negative, we take d = (2 + √29) / 2
Substituting this value of d in the equation a+2d = 6, we get:
a = 6 - 2d = 6 - (2 + √29) = 4 - √29
Therefore, the third term of the AP is a+2d = (4 - √29) + 2((2 + √29) / 2) = 6 - √29 ≈ 4.31
Hence, the correct option is (B) 6.
Let the first term of the AP be a and the common difference be d.
Then the five numbers are: a, a+d, a+2d, a+3d, a+4d
Given that the sum of the five numbers is 30:
a + a+d + a+2d + a+3d + a+4d = 30
5a + 10d = 30
a + 2d = 6
Also given that the sum of their squares is 220:
a^2 + (a+d)^2 + (a+2d)^2 + (a+3d)^2 + (a+4d)^2 = 220
5a^2 + 30d^2 + 20ad = 220
a^2 + 6d^2 + 4ad = 44
Substituting a+2d=6, we get:
a = 6 - 2d
Substituting this value of a in the above equation, we get:
(6-2d)^2 + 6d^2 + 4(6-2d)d = 44
Simplifying this equation, we get:
4d^2 - 8d - 5 = 0
On solving this quadratic equation, we get:
d = (-(-8) ± √((-8)^2 - 4(4)(-5))) / (2(4))
d = (8 ± √116) / 8
d = (2 ± √29) / 2
Since the common difference of an AP cannot be negative, we take d = (2 + √29) / 2
Substituting this value of d in the equation a+2d = 6, we get:
a = 6 - 2d = 6 - (2 + √29) = 4 - √29
Therefore, the third term of the AP is a+2d = (4 - √29) + 2((2 + √29) / 2) = 6 - √29 ≈ 4.31
Hence, the correct option is (B) 6.
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