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If the arithmetic progression whose common difference is non zero the sum of first 3n term is equal to sum of next n terms . Then find the ratio of the sum of first 2n terms to the sum of next 2n terms.?
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If the arithmetic progression whose common difference is non zero the ...
**Solution:**
Let's assume the first term of the arithmetic progression is 'a' and the common difference is 'd'.
The sum of the first 3n terms can be represented as:
S1 = n/2 * (2a + (3n - 1) * d)
The sum of the next n terms can be represented as:
S2 = n/2 * (2(a + (3n - 1)d) + (n - 1) * d)
According to the given condition, S1 = S2.
n/2 * (2a + (3n - 1) * d) = n/2 * (2(a + (3n - 1)d) + (n - 1) * d)
Cancelling n/2 from both sides:
2a + (3n - 1) * d = 2(a + (3n - 1)d) + (n - 1) * d
2a + 3nd - d = 2a + 6nd - 2d + nd - d
Rearranging the terms:
2a - 2a + 3nd - 6nd - nd + 2d + d = 0
4d + nd - 3nd = 0
Factoring out 'd':
d(4 + n - 3n) = 0
Since the common difference is non-zero, we can conclude that:
4 + n - 3n = 0
-2n + 4 = 0
-2n = -4
n = 2
Therefore, the ratio of the sum of the first 2n terms to the sum of the next 2n terms is:
S1/S2 = 2n/2n = 2/2 = 1
The ratio is equal to 1, which implies that the sum of the first 2n terms is equal to the sum of the next 2n terms.
Let's assume the first term of the arithmetic progression is 'a' and the common difference is 'd'.
The sum of the first 3n terms can be represented as:
S1 = n/2 * (2a + (3n - 1) * d)
The sum of the next n terms can be represented as:
S2 = n/2 * (2(a + (3n - 1)d) + (n - 1) * d)
According to the given condition, S1 = S2.
n/2 * (2a + (3n - 1) * d) = n/2 * (2(a + (3n - 1)d) + (n - 1) * d)
Cancelling n/2 from both sides:
2a + (3n - 1) * d = 2(a + (3n - 1)d) + (n - 1) * d
2a + 3nd - d = 2a + 6nd - 2d + nd - d
Rearranging the terms:
2a - 2a + 3nd - 6nd - nd + 2d + d = 0
4d + nd - 3nd = 0
Factoring out 'd':
d(4 + n - 3n) = 0
Since the common difference is non-zero, we can conclude that:
4 + n - 3n = 0
-2n + 4 = 0
-2n = -4
n = 2
Therefore, the ratio of the sum of the first 2n terms to the sum of the next 2n terms is:
S1/S2 = 2n/2n = 2/2 = 1
The ratio is equal to 1, which implies that the sum of the first 2n terms is equal to the sum of the next 2n terms.
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If the arithmetic progression whose common difference is non zero the ...
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If the arithmetic progression whose common difference is non zero the sum of first 3n term is equal to sum of next n terms . Then find the ratio of the sum of first 2n terms to the sum of next 2n terms.?
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If the arithmetic progression whose common difference is non zero the sum of first 3n term is equal to sum of next n terms . Then find the ratio of the sum of first 2n terms to the sum of next 2n terms.? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the arithmetic progression whose common difference is non zero the sum of first 3n term is equal to sum of next n terms . Then find the ratio of the sum of first 2n terms to the sum of next 2n terms.? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the arithmetic progression whose common difference is non zero the sum of first 3n term is equal to sum of next n terms . Then find the ratio of the sum of first 2n terms to the sum of next 2n terms.?.
If the arithmetic progression whose common difference is non zero the sum of first 3n term is equal to sum of next n terms . Then find the ratio of the sum of first 2n terms to the sum of next 2n terms.? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the arithmetic progression whose common difference is non zero the sum of first 3n term is equal to sum of next n terms . Then find the ratio of the sum of first 2n terms to the sum of next 2n terms.? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the arithmetic progression whose common difference is non zero the sum of first 3n term is equal to sum of next n terms . Then find the ratio of the sum of first 2n terms to the sum of next 2n terms.?.
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