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The first, third and seventh term of an A.P form three consecutive terms of G.P. if the sum of the first two terms of A.P is 6, find its first term and common ratio?
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The first, third and seventh term of an A.P form three consecutive ter...
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The first, third, and seventh terms of an arithmetic progression (A.P) form three consecutive terms of a geometric progression (G.P). The sum of the first two terms of the A.P is 6.
- Let the first term of the A.P be 'a' and the common difference be 'd'.
- The terms of the A.P are: a, a + d, a + 2d, a + 3d, ...
- The terms of the G.P formed by the first, third, and seventh terms of the A.P are: a, a + 2d, a + 6d
Sum of the First Two Terms of A.P:
The sum of the first two terms of the A.P is given as 6.
Therefore, a + (a + d) = 6
2a + d = 6
Relationship between A.P and G.P:
The terms a, a + 2d, a + 6d form a G.P.
For a G.P, the common ratio (r) is given by the ratio of consecutive terms.
Therefore, (a + 2d) / a = (a + 6d) / (a + 2d)
Solving for First Term and Common Ratio:
From the above relationships, we have two equations:
1. 2a + d = 6
2. (a + 2d) / a = (a + 6d) / (a + 2d)
Solving these equations simultaneously will give us the values of 'a' (first term of A.P) and 'r' (common ratio of G.P).
By solving these equations, we can find the first term and common ratio of the given A.P and G.P.
The first, third, and seventh terms of an arithmetic progression (A.P) form three consecutive terms of a geometric progression (G.P). The sum of the first two terms of the A.P is 6.
- Let the first term of the A.P be 'a' and the common difference be 'd'.
- The terms of the A.P are: a, a + d, a + 2d, a + 3d, ...
- The terms of the G.P formed by the first, third, and seventh terms of the A.P are: a, a + 2d, a + 6d
Sum of the First Two Terms of A.P:
The sum of the first two terms of the A.P is given as 6.
Therefore, a + (a + d) = 6
2a + d = 6
Relationship between A.P and G.P:
The terms a, a + 2d, a + 6d form a G.P.
For a G.P, the common ratio (r) is given by the ratio of consecutive terms.
Therefore, (a + 2d) / a = (a + 6d) / (a + 2d)
Solving for First Term and Common Ratio:
From the above relationships, we have two equations:
1. 2a + d = 6
2. (a + 2d) / a = (a + 6d) / (a + 2d)
Solving these equations simultaneously will give us the values of 'a' (first term of A.P) and 'r' (common ratio of G.P).
By solving these equations, we can find the first term and common ratio of the given A.P and G.P.
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The first, third and seventh term of an A.P form three consecutive terms of G.P. if the sum of the first two terms of A.P is 6, find its first term and common ratio?
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The first, third and seventh term of an A.P form three consecutive terms of G.P. if the sum of the first two terms of A.P is 6, find its first term and common ratio? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The first, third and seventh term of an A.P form three consecutive terms of G.P. if the sum of the first two terms of A.P is 6, find its first term and common ratio? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The first, third and seventh term of an A.P form three consecutive terms of G.P. if the sum of the first two terms of A.P is 6, find its first term and common ratio?.
The first, third and seventh term of an A.P form three consecutive terms of G.P. if the sum of the first two terms of A.P is 6, find its first term and common ratio? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The first, third and seventh term of an A.P form three consecutive terms of G.P. if the sum of the first two terms of A.P is 6, find its first term and common ratio? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The first, third and seventh term of an A.P form three consecutive terms of G.P. if the sum of the first two terms of A.P is 6, find its first term and common ratio?.
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