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The first term of an arithmetic progression is 2 and the fourth term is 6. If the sum of first n terms of the progression is 6800, find the value of n.
- a)100
- b)120
- c)150
- d)160
Correct answer is option 'A'. Can you explain this answer?
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The first term of an arithmetic progression is 2 and the fourth term i...
Given:
- The first term of an arithmetic progression is 2.
- The fourth term of the arithmetic progression is 6.
- The sum of the first n terms of the progression is 6800.
To find:
The value of n.
Solution:
Let's denote the common difference of the arithmetic progression as d.
Finding the value of the common difference (d):
The fourth term of the arithmetic progression can be expressed as:
a + 3d = 6 ...(1) [where a is the first term]
Substituting the value of the first term (a) as 2 in equation (1), we get:
2 + 3d = 6
Simplifying the equation, we have:
3d = 4
d = 4/3
So, the common difference (d) is 4/3.
Finding the value of n:
The sum of the first n terms of an arithmetic progression can be expressed as:
Sn = (n/2)(2a + (n-1)d) ...(2)
Given that the sum of the first n terms (Sn) is 6800, we can substitute the values in equation (2):
6800 = (n/2)(2(2) + (n-1)(4/3))
Simplifying the equation, we have:
6800 = (n/2)(4 + (4/3)(n-1))
Multiplying both sides of the equation by 2 to eliminate the fraction, we get:
13600 = n(4 + (4/3)(n-1))
Expanding the equation, we have:
13600 = 4n + (4/3)n^2 - 4/3n
Multiplying both sides of the equation by 3 to eliminate the fraction, we get:
40800 = 12n + 4n^2 - 4n
Rearranging the equation to form a quadratic equation, we have:
4n^2 + 8n - 40800 = 0
Simplifying the equation, we have:
n^2 + 2n - 10200 = 0
Factoring the quadratic equation, we get:
(n + 102)(n - 100) = 0
Therefore, n = -102 or n = 100.
Since the number of terms (n) cannot be negative, the value of n is 100.
Conclusion:
The value of n is 100. Therefore, the correct answer is option A.
- The first term of an arithmetic progression is 2.
- The fourth term of the arithmetic progression is 6.
- The sum of the first n terms of the progression is 6800.
To find:
The value of n.
Solution:
Let's denote the common difference of the arithmetic progression as d.
Finding the value of the common difference (d):
The fourth term of the arithmetic progression can be expressed as:
a + 3d = 6 ...(1) [where a is the first term]
Substituting the value of the first term (a) as 2 in equation (1), we get:
2 + 3d = 6
Simplifying the equation, we have:
3d = 4
d = 4/3
So, the common difference (d) is 4/3.
Finding the value of n:
The sum of the first n terms of an arithmetic progression can be expressed as:
Sn = (n/2)(2a + (n-1)d) ...(2)
Given that the sum of the first n terms (Sn) is 6800, we can substitute the values in equation (2):
6800 = (n/2)(2(2) + (n-1)(4/3))
Simplifying the equation, we have:
6800 = (n/2)(4 + (4/3)(n-1))
Multiplying both sides of the equation by 2 to eliminate the fraction, we get:
13600 = n(4 + (4/3)(n-1))
Expanding the equation, we have:
13600 = 4n + (4/3)n^2 - 4/3n
Multiplying both sides of the equation by 3 to eliminate the fraction, we get:
40800 = 12n + 4n^2 - 4n
Rearranging the equation to form a quadratic equation, we have:
4n^2 + 8n - 40800 = 0
Simplifying the equation, we have:
n^2 + 2n - 10200 = 0
Factoring the quadratic equation, we get:
(n + 102)(n - 100) = 0
Therefore, n = -102 or n = 100.
Since the number of terms (n) cannot be negative, the value of n is 100.
Conclusion:
The value of n is 100. Therefore, the correct answer is option A.
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The first term of an arithmetic progression is 2 and the fourth term is 6. If the sum of first n terms of the progression is 6800, find the value of n.a)100b)120c)150d)160Correct answer is option 'A'. Can you explain this answer?
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The first term of an arithmetic progression is 2 and the fourth term is 6. If the sum of first n terms of the progression is 6800, find the value of n.a)100b)120c)150d)160Correct answer is option 'A'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The first term of an arithmetic progression is 2 and the fourth term is 6. If the sum of first n terms of the progression is 6800, find the value of n.a)100b)120c)150d)160Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The first term of an arithmetic progression is 2 and the fourth term is 6. If the sum of first n terms of the progression is 6800, find the value of n.a)100b)120c)150d)160Correct answer is option 'A'. Can you explain this answer?.
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