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The sum of the 5th term and the 18th term of an arithmetic progression is equal to the sum of the 7th, 12th and the 15th term of the same progression. Which element of the series would necessarily be equal to zero?
- a)8th
- b)9th
- c)11th
- d)10th
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
The sum of the 5th term and the 18th term of an arithmetic progression...
Let the 1st term of the arithmetic progression be a and the common difference be d.
The 5th term will be (a + 4d) and the 18th term will be (a + 17d).
Also, the 7th term will be (a + 6d), the 12th term will be (a + 11d) and the 15th term will be (a +14d).
Now, from the given condition, we get (a + 4 d) + (a + 17d) = (a + d) + (a + 11d + (a + 14d)
Also, the 7th term will be (a + 6d), the 12th term will be (a + 11d) and the 15th term will be (a +14d).
Now, from the given condition, we get (a + 4 d) + (a + 17d) = (a + d) + (a + 11d + (a + 14d)
2a + 21d = 3a + 31d
a + 10d = 0 (a + 10d) is the 11th term of the series.
Most Upvoted Answer
The sum of the 5th term and the 18th term of an arithmetic progression...
Given:
- The arithmetic progression has a common difference.
- The sum of the 5th term and the 18th term is equal to the sum of the 7th, 12th, and 15th term.
To find:
Which element of the series would necessarily be equal to zero?
Solution:
Let's assume the first term of the arithmetic progression is 'a' and the common difference is 'd'.
The general formula for the nth term of an arithmetic progression is given by:
an = a + (n-1)d
Step 1: Finding the terms using the given information
- The 5th term: a + 4d
- The 18th term: a + 17d
- The 7th term: a + 6d
- The 12th term: a + 11d
- The 15th term: a + 14d
Step 2: Writing the equation using the given information
According to the given information, the sum of the 5th term and the 18th term is equal to the sum of the 7th, 12th, and 15th term. Mathematically, this can be represented as:
(a + 4d) + (a + 17d) = (a + 6d) + (a + 11d) + (a + 14d)
Step 3: Simplifying the equation
a + 4d + a + 17d = a + 6d + a + 11d + a + 14d
2a + 21d = 4a + 31d
2a - 4a = 31d - 21d
-2a = 10d
a = -5d/1
This means that the first term 'a' is equal to -5d.
Step 4: Identifying the element that is necessarily equal to zero
Now, we need to find the element of the series that is necessarily equal to zero. Let's substitute the value of 'a' in terms of 'd' into the formula for the nth term:
an = a + (n-1)d
Substituting a = -5d into the formula:
an = -5d + (n-1)d
an = -5d + nd - d
an = (n-6)d
To find the element that is necessarily equal to zero, we need to solve the equation (n-6)d = 0.
Step 5: Solving the equation
(n-6)d = 0
Either (n-6) = 0, which implies n = 6
Or d = 0 (common difference)
Since 'd' represents the common difference and it cannot be zero (as it is an arithmetic progression), the only possible solution is n = 6.
Therefore, the 6th element (11th term) of the series would necessarily be equal to zero.
Hence, the correct answer is option 'C' - 11th term.
- The arithmetic progression has a common difference.
- The sum of the 5th term and the 18th term is equal to the sum of the 7th, 12th, and 15th term.
To find:
Which element of the series would necessarily be equal to zero?
Solution:
Let's assume the first term of the arithmetic progression is 'a' and the common difference is 'd'.
The general formula for the nth term of an arithmetic progression is given by:
an = a + (n-1)d
Step 1: Finding the terms using the given information
- The 5th term: a + 4d
- The 18th term: a + 17d
- The 7th term: a + 6d
- The 12th term: a + 11d
- The 15th term: a + 14d
Step 2: Writing the equation using the given information
According to the given information, the sum of the 5th term and the 18th term is equal to the sum of the 7th, 12th, and 15th term. Mathematically, this can be represented as:
(a + 4d) + (a + 17d) = (a + 6d) + (a + 11d) + (a + 14d)
Step 3: Simplifying the equation
a + 4d + a + 17d = a + 6d + a + 11d + a + 14d
2a + 21d = 4a + 31d
2a - 4a = 31d - 21d
-2a = 10d
a = -5d/1
This means that the first term 'a' is equal to -5d.
Step 4: Identifying the element that is necessarily equal to zero
Now, we need to find the element of the series that is necessarily equal to zero. Let's substitute the value of 'a' in terms of 'd' into the formula for the nth term:
an = a + (n-1)d
Substituting a = -5d into the formula:
an = -5d + (n-1)d
an = -5d + nd - d
an = (n-6)d
To find the element that is necessarily equal to zero, we need to solve the equation (n-6)d = 0.
Step 5: Solving the equation
(n-6)d = 0
Either (n-6) = 0, which implies n = 6
Or d = 0 (common difference)
Since 'd' represents the common difference and it cannot be zero (as it is an arithmetic progression), the only possible solution is n = 6.
Therefore, the 6th element (11th term) of the series would necessarily be equal to zero.
Hence, the correct answer is option 'C' - 11th term.
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The sum of the 5th term and the 18th term of an arithmetic progression is equal to the sum of the 7th, 12th and the 15th term of the same progression. Which element of the series would necessarily be equal to zero?a)8thb)9thc)11thd)10thCorrect answer is option 'C'. Can you explain this answer?
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