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The sum of the first two terms of a G.P. is 5/3 and the sum to infinity of the series is 3. The common ratio is
- a)1/3
- b)2/3
- c)– 2/3
- d)none of these
Correct answer is option 'B,C'. Can you explain this answer?
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Verified Answer
The sum of the first two terms of a G.P. is 5/3 and the sum to infinit...
The first two terms = a, ar a + ar = 5/3
a/(1 – r)= 3 ⇒ a = 3(1-r)a(1 + r)=5/3
⇒3(1-r)(1 + r)=5/3
⇒1 – r^2 = 5/9
=> r^2 = 4/9
=> Common ratio, r = +2/3 or -2/3
Most Upvoted Answer
The sum of the first two terms of a G.P. is 5/3 and the sum to infinit...
Given: sum of first two terms of G.P. is 5/3 and sum to infinity is 3.
To find: Common ratio of the G.P.
Solution:
Let the first term of the G.P. be 'a' and the common ratio be 'r'.
According to the given information,
a + ar = 5/3 ----(1)
The sum to infinity of a G.P. is given by the formula:
Sum to infinity = a/(1-r)
3 = a/(1-r) ----(2)
Solving equations (1) and (2), we get:
a = 5/9 and r = 2/3 or r = 1/3
Therefore, the common ratio of the G.P. can be either 2/3 or 1/3.
Hence, the correct answer is options B and C.
To find: Common ratio of the G.P.
Solution:
Let the first term of the G.P. be 'a' and the common ratio be 'r'.
According to the given information,
a + ar = 5/3 ----(1)
The sum to infinity of a G.P. is given by the formula:
Sum to infinity = a/(1-r)
3 = a/(1-r) ----(2)
Solving equations (1) and (2), we get:
a = 5/9 and r = 2/3 or r = 1/3
Therefore, the common ratio of the G.P. can be either 2/3 or 1/3.
Hence, the correct answer is options B and C.
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Community Answer
The sum of the first two terms of a G.P. is 5/3 and the sum to infinit...
Given information:
- The sum of the first two terms of a G.P. is 5/3
- The sum to infinity of the series is 3
To find: The common ratio of the G.P.
Solution:
Let the first term of the G.P. be 'a' and the common ratio be 'r'. Then, the sum of the first two terms of the G.P. is given by:
a + ar = a(1+r)
We are given that this sum is equal to 5/3. Therefore, we have:
a(1+r) = 5/3
Also, the sum to infinity of the G.P. is given by:
S = a/(1-r)
We are given that S = 3. Therefore, we have:
a/(1-r) = 3
Solving these two equations simultaneously, we get:
a = 5/6 and r = 2/3 or -1
Therefore, the common ratio of the G.P. can be either 2/3 or -1.
But we know that a G.P. with a negative common ratio cannot have a finite sum to infinity, since the terms of the G.P. will alternate in sign and their absolute values will not tend to zero. Therefore, the common ratio of the G.P. must be positive, which means that the common ratio is 2/3.
Therefore, the correct answer is option (b) 2/3.
- The sum of the first two terms of a G.P. is 5/3
- The sum to infinity of the series is 3
To find: The common ratio of the G.P.
Solution:
Let the first term of the G.P. be 'a' and the common ratio be 'r'. Then, the sum of the first two terms of the G.P. is given by:
a + ar = a(1+r)
We are given that this sum is equal to 5/3. Therefore, we have:
a(1+r) = 5/3
Also, the sum to infinity of the G.P. is given by:
S = a/(1-r)
We are given that S = 3. Therefore, we have:
a/(1-r) = 3
Solving these two equations simultaneously, we get:
a = 5/6 and r = 2/3 or -1
Therefore, the common ratio of the G.P. can be either 2/3 or -1.
But we know that a G.P. with a negative common ratio cannot have a finite sum to infinity, since the terms of the G.P. will alternate in sign and their absolute values will not tend to zero. Therefore, the common ratio of the G.P. must be positive, which means that the common ratio is 2/3.
Therefore, the correct answer is option (b) 2/3.
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