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In an arithmetic series consisting of 51 terms, the sum of the first three terms is 65 and the sum of the middle three terms is 129. What is the first term and the common difference of the series?
- a)64, 9/8
- b)32, 8/9
- c)187/9, 8/9
- d)72, 9/8
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
In an arithmetic series consisting of 51 terms, the sum of the first t...
Given a + (a + d) + (a + 2d)= 65
=> 3a + 3d =65…. (1)
The middle terms are 25th , 26th and 27th terms
=> a + 24d + a + 25d + a + 26d = 129
=> 3a + 75d
129 ….(2)
From (1) and (2) d= 8/9 =>a= 187/9
Most Upvoted Answer
In an arithmetic series consisting of 51 terms, the sum of the first t...
Given information:
- The arithmetic series consists of 51 terms.
- The sum of the first three terms is 65.
- The sum of the middle three terms is 129.
Let's find the first term:
- The sum of the first three terms can be expressed as:
S3 = 3/2 * a + 3/2 * d, where a is the first term and d is the common difference.
- We are given that S3 = 65, so we can write the equation as:
65 = 3/2 * a + 3/2 * d
Let's find the sum of the middle three terms:
- The sum of the middle three terms can be expressed as:
S_middle = 3/2 * (a + 2d) + 3/2 * (a + d) + 3/2 * a
- We are given that S_middle = 129, so we can write the equation as:
129 = 3/2 * (a + 2d) + 3/2 * (a + d) + 3/2 * a
Solving the equations:
1. Rewrite the equations:
65 = 3/2 * a + 3/2 * d
129 = 3/2 * (a + 2d) + 3/2 * (a + d) + 3/2 * a
2. Simplify the equations:
65 = 3/2 * (2a + d)
129 = 3/2 * (3a + 3d)
3. Remove the fractions by multiplying both sides of the equations by 2:
130 = 3(2a + d)
258 = 3(3a + 3d)
4. Simplify the equations:
130 = 6a + 3d
258 = 9a + 9d
5. Rearrange the first equation to solve for d:
3d = 130 - 6a
d = (130 - 6a)/3
6. Substitute the value of d in the second equation:
258 = 9a + 9((130 - 6a)/3)
258 = 9a + 3(130 - 6a)
258 = 9a + 390 - 18a
258 - 390 = -9a
-132 = -9a
a = -132/-9
a = 187/9
Conclusion:
The first term (a) of the arithmetic series is 187/9 and the common difference (d) is (130 - 6a)/3, which simplifies to 8/9. Therefore, the correct answer is option C: 187/9, 8/9.
- The arithmetic series consists of 51 terms.
- The sum of the first three terms is 65.
- The sum of the middle three terms is 129.
Let's find the first term:
- The sum of the first three terms can be expressed as:
S3 = 3/2 * a + 3/2 * d, where a is the first term and d is the common difference.
- We are given that S3 = 65, so we can write the equation as:
65 = 3/2 * a + 3/2 * d
Let's find the sum of the middle three terms:
- The sum of the middle three terms can be expressed as:
S_middle = 3/2 * (a + 2d) + 3/2 * (a + d) + 3/2 * a
- We are given that S_middle = 129, so we can write the equation as:
129 = 3/2 * (a + 2d) + 3/2 * (a + d) + 3/2 * a
Solving the equations:
1. Rewrite the equations:
65 = 3/2 * a + 3/2 * d
129 = 3/2 * (a + 2d) + 3/2 * (a + d) + 3/2 * a
2. Simplify the equations:
65 = 3/2 * (2a + d)
129 = 3/2 * (3a + 3d)
3. Remove the fractions by multiplying both sides of the equations by 2:
130 = 3(2a + d)
258 = 3(3a + 3d)
4. Simplify the equations:
130 = 6a + 3d
258 = 9a + 9d
5. Rearrange the first equation to solve for d:
3d = 130 - 6a
d = (130 - 6a)/3
6. Substitute the value of d in the second equation:
258 = 9a + 9((130 - 6a)/3)
258 = 9a + 3(130 - 6a)
258 = 9a + 390 - 18a
258 - 390 = -9a
-132 = -9a
a = -132/-9
a = 187/9
Conclusion:
The first term (a) of the arithmetic series is 187/9 and the common difference (d) is (130 - 6a)/3, which simplifies to 8/9. Therefore, the correct answer is option C: 187/9, 8/9.
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In an arithmetic series consisting of 51 terms, the sum of the first three terms is 65 and the sum of the middle three terms is 129. What is the first term and the common difference of the series?a)64, 9/8b)32, 8/9c)187/9, 8/9d)72, 9/8Correct answer is option 'C'. Can you explain this answer?
Question Description
In an arithmetic series consisting of 51 terms, the sum of the first three terms is 65 and the sum of the middle three terms is 129. What is the first term and the common difference of the series?a)64, 9/8b)32, 8/9c)187/9, 8/9d)72, 9/8Correct answer is option 'C'. Can you explain this answer? for Quant 2024 is part of Quant preparation. The Question and answers have been prepared according to the Quant exam syllabus. Information about In an arithmetic series consisting of 51 terms, the sum of the first three terms is 65 and the sum of the middle three terms is 129. What is the first term and the common difference of the series?a)64, 9/8b)32, 8/9c)187/9, 8/9d)72, 9/8Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Quant 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In an arithmetic series consisting of 51 terms, the sum of the first three terms is 65 and the sum of the middle three terms is 129. What is the first term and the common difference of the series?a)64, 9/8b)32, 8/9c)187/9, 8/9d)72, 9/8Correct answer is option 'C'. Can you explain this answer?.
In an arithmetic series consisting of 51 terms, the sum of the first three terms is 65 and the sum of the middle three terms is 129. What is the first term and the common difference of the series?a)64, 9/8b)32, 8/9c)187/9, 8/9d)72, 9/8Correct answer is option 'C'. Can you explain this answer? for Quant 2024 is part of Quant preparation. The Question and answers have been prepared according to the Quant exam syllabus. Information about In an arithmetic series consisting of 51 terms, the sum of the first three terms is 65 and the sum of the middle three terms is 129. What is the first term and the common difference of the series?a)64, 9/8b)32, 8/9c)187/9, 8/9d)72, 9/8Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Quant 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In an arithmetic series consisting of 51 terms, the sum of the first three terms is 65 and the sum of the middle three terms is 129. What is the first term and the common difference of the series?a)64, 9/8b)32, 8/9c)187/9, 8/9d)72, 9/8Correct answer is option 'C'. Can you explain this answer?.
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In an arithmetic series consisting of 51 terms, the sum of the first three terms is 65 and the sum of the middle three terms is 129. What is the first term and the common difference of the series?a)64, 9/8b)32, 8/9c)187/9, 8/9d)72, 9/8Correct answer is option 'C'. Can you explain this answer?, a detailed solution for In an arithmetic series consisting of 51 terms, the sum of the first three terms is 65 and the sum of the middle three terms is 129. What is the first term and the common difference of the series?a)64, 9/8b)32, 8/9c)187/9, 8/9d)72, 9/8Correct answer is option 'C'. Can you explain this answer? has been provided alongside types of In an arithmetic series consisting of 51 terms, the sum of the first three terms is 65 and the sum of the middle three terms is 129. What is the first term and the common difference of the series?a)64, 9/8b)32, 8/9c)187/9, 8/9d)72, 9/8Correct answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
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