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Consider an arithmetic progression with n terms. If the common difference is increased by 1, then nth term increases by 19. If the 5th term of the progression is 28 and the average of the first and last terms is 61, then the 10th term of the progression is:
- a)54
- b)56
- c)58
- d)60
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Consider an arithmetic progression with n terms. If the common differe...
an = a + (n – 1)d
If d increased to d + 1
a1n = an + 19
a + (n – 1)(d + 1) = a + (n – 1)d + 19
(n – 1)d + n – 1 = (n – 1)d + 19
n = 20
a5 = 28
a + 4d = 28 …(i)
2a + 19d = 122 …(ii)
From equation (i) and (ii)
2a + 19d = 122
d = 6
a = 4
a10 = a + 9d
= 4 + 54
= 58
If d increased to d + 1
a1n = an + 19
a + (n – 1)(d + 1) = a + (n – 1)d + 19
(n – 1)d + n – 1 = (n – 1)d + 19
n = 20
a5 = 28
a + 4d = 28 …(i)
2a + 19d = 122 …(ii)
From equation (i) and (ii)
2a + 19d = 122
d = 6
a = 4
a10 = a + 9d
= 4 + 54
= 58
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Consider an arithmetic progression with n terms. If the common differe...
Given information:
- The common difference of the arithmetic progression is initially d.
- When the common difference is increased by 1, the nth term of the progression increases by 19.
- The 5th term of the progression is 28.
- The average of the first and last terms is 61.
Approach:
1. Find the initial common difference (d) using the 5th term of the progression.
2. Find the first term (a) of the progression using the average of the first and last terms.
3. Use the formula for the nth term of an arithmetic progression to find the value of n.
4. Use the formula for the nth term of an arithmetic progression to find the 10th term.
Solution:
1. Finding the initial common difference (d):
The 5th term of the progression is given as 28.
We know that the nth term of an arithmetic progression can be represented as:
Tn = a + (n-1)d
where Tn is the nth term, a is the first term, and d is the common difference.
Substituting the given values:
28 = a + 4d ...(1)
2. Finding the first term (a):
The average of the first and last terms is given as 61.
The average of two numbers can be calculated by taking their sum and dividing it by 2.
So, we have:
(a + a + (n-1)d)/2 = 61
2a + (n-1)d = 122 ...(2)
3. Finding the value of n:
We can substitute the value of a from equation (1) into equation (2):
2(a + 4d) + (n-1)d = 122
2a + 8d + (n-1)d = 122
Substituting the value of a from equation (1):
2(28) + 8d + (n-1)d = 122
56 + 8d + nd - d = 122
56 + 7d + nd = 122
7d + nd = 66 ...(3)
4. Finding the 10th term:
We need to find the value of the 10th term, which can be represented as:
T10 = a + 9d
Substituting the value of a from equation (1):
T10 = (28 - 4d) + 9d
T10 = 28 + 5d ...(4)
Now, we are given that when the common difference is increased by 1, the nth term increases by 19.
So, we have:
7d + nd = 66 ...(3)
7(d+1) + n(d+1) = 66 + 19
7d + 7 + nd + n = 85
7d + nd + n = 78 ...(5)
We can solve equations (3) and (5) simultaneously to find the values of d and n.
Subtracting equation (5) from equation (3), we get:
6d = 12
d = 2
Substituting the value of d = 2 in equation (5), we get:
14 + 2n + n = 78
3n = 64
n = 64/
- The common difference of the arithmetic progression is initially d.
- When the common difference is increased by 1, the nth term of the progression increases by 19.
- The 5th term of the progression is 28.
- The average of the first and last terms is 61.
Approach:
1. Find the initial common difference (d) using the 5th term of the progression.
2. Find the first term (a) of the progression using the average of the first and last terms.
3. Use the formula for the nth term of an arithmetic progression to find the value of n.
4. Use the formula for the nth term of an arithmetic progression to find the 10th term.
Solution:
1. Finding the initial common difference (d):
The 5th term of the progression is given as 28.
We know that the nth term of an arithmetic progression can be represented as:
Tn = a + (n-1)d
where Tn is the nth term, a is the first term, and d is the common difference.
Substituting the given values:
28 = a + 4d ...(1)
2. Finding the first term (a):
The average of the first and last terms is given as 61.
The average of two numbers can be calculated by taking their sum and dividing it by 2.
So, we have:
(a + a + (n-1)d)/2 = 61
2a + (n-1)d = 122 ...(2)
3. Finding the value of n:
We can substitute the value of a from equation (1) into equation (2):
2(a + 4d) + (n-1)d = 122
2a + 8d + (n-1)d = 122
Substituting the value of a from equation (1):
2(28) + 8d + (n-1)d = 122
56 + 8d + nd - d = 122
56 + 7d + nd = 122
7d + nd = 66 ...(3)
4. Finding the 10th term:
We need to find the value of the 10th term, which can be represented as:
T10 = a + 9d
Substituting the value of a from equation (1):
T10 = (28 - 4d) + 9d
T10 = 28 + 5d ...(4)
Now, we are given that when the common difference is increased by 1, the nth term increases by 19.
So, we have:
7d + nd = 66 ...(3)
7(d+1) + n(d+1) = 66 + 19
7d + 7 + nd + n = 85
7d + nd + n = 78 ...(5)
We can solve equations (3) and (5) simultaneously to find the values of d and n.
Subtracting equation (5) from equation (3), we get:
6d = 12
d = 2
Substituting the value of d = 2 in equation (5), we get:
14 + 2n + n = 78
3n = 64
n = 64/
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Consider an arithmetic progression with n terms. If the common difference is increased by 1, then nth term increases by 19. If the 5th term of the progression is 28 and the average of the first and last terms is 61, then the 10th term of the progression is:a)54b)56c)58d)60Correct answer is option 'C'. Can you explain this answer?
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Consider an arithmetic progression with n terms. If the common difference is increased by 1, then nth term increases by 19. If the 5th term of the progression is 28 and the average of the first and last terms is 61, then the 10th term of the progression is:a)54b)56c)58d)60Correct answer is option 'C'. Can you explain this answer? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Consider an arithmetic progression with n terms. If the common difference is increased by 1, then nth term increases by 19. If the 5th term of the progression is 28 and the average of the first and last terms is 61, then the 10th term of the progression is:a)54b)56c)58d)60Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider an arithmetic progression with n terms. If the common difference is increased by 1, then nth term increases by 19. If the 5th term of the progression is 28 and the average of the first and last terms is 61, then the 10th term of the progression is:a)54b)56c)58d)60Correct answer is option 'C'. Can you explain this answer?.
Consider an arithmetic progression with n terms. If the common difference is increased by 1, then nth term increases by 19. If the 5th term of the progression is 28 and the average of the first and last terms is 61, then the 10th term of the progression is:a)54b)56c)58d)60Correct answer is option 'C'. Can you explain this answer? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Consider an arithmetic progression with n terms. If the common difference is increased by 1, then nth term increases by 19. If the 5th term of the progression is 28 and the average of the first and last terms is 61, then the 10th term of the progression is:a)54b)56c)58d)60Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider an arithmetic progression with n terms. If the common difference is increased by 1, then nth term increases by 19. If the 5th term of the progression is 28 and the average of the first and last terms is 61, then the 10th term of the progression is:a)54b)56c)58d)60Correct answer is option 'C'. Can you explain this answer?.
Solutions for Consider an arithmetic progression with n terms. If the common difference is increased by 1, then nth term increases by 19. If the 5th term of the progression is 28 and the average of the first and last terms is 61, then the 10th term of the progression is:a)54b)56c)58d)60Correct answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Class 10.
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