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The sum of the first two term s of a GP is 5/3 and the sum of infinity of the series is 3. The common ratio is
- a)1/3
- b)2/3
- c)-1/3
- d)none of these
Correct answer is option 'D'. Can you explain this answer?
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The sum of the first two term s of a GP is 5/3 and the sum of infinity...
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The sum of the first two term s of a GP is 5/3 and the sum of infinity...
Given:
Sum of first two terms of GP, S2 = 5/3
Sum of infinite terms of GP, S∞ = 3
To find: Common ratio of the GP
Solution:
Let the first term of the GP be a and the common ratio be r.
Then, the second term of the GP is ar.
Using the formula for sum of first two terms of a GP,
S2 = a + ar = a(1 + r) = 5/3
Similarly, using the formula for sum of infinite terms of a GP,
S∞ = a/(1-r) = 3
We can now solve these two equations to find the values of a and r.
a(1 + r) = 5/3 ...(1)
a/(1-r) = 3 ...(2)
From equation (2), we get:
a = 3(1-r)
Substituting this value of a in equation (1), we get:
3(1-r)(1+r) = 5/3
9(1-r^2) = 5
1-r^2 = 5/9
r^2 = 4/9
r = ±2/3
Since r cannot be negative (as it is a geometric progression), we take the positive value of r:
r = 2/3
Therefore, the common ratio of the GP is 2/3.
Hence, the correct option is (d) none of these.
Sum of first two terms of GP, S2 = 5/3
Sum of infinite terms of GP, S∞ = 3
To find: Common ratio of the GP
Solution:
Let the first term of the GP be a and the common ratio be r.
Then, the second term of the GP is ar.
Using the formula for sum of first two terms of a GP,
S2 = a + ar = a(1 + r) = 5/3
Similarly, using the formula for sum of infinite terms of a GP,
S∞ = a/(1-r) = 3
We can now solve these two equations to find the values of a and r.
a(1 + r) = 5/3 ...(1)
a/(1-r) = 3 ...(2)
From equation (2), we get:
a = 3(1-r)
Substituting this value of a in equation (1), we get:
3(1-r)(1+r) = 5/3
9(1-r^2) = 5
1-r^2 = 5/9
r^2 = 4/9
r = ±2/3
Since r cannot be negative (as it is a geometric progression), we take the positive value of r:
r = 2/3
Therefore, the common ratio of the GP is 2/3.
Hence, the correct option is (d) none of these.
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The sum of the first two term s of a GP is 5/3 and the sum of infinity of the series is 3. The common ratio isa)1/3b)2/3c)-1/3d)none of theseCorrect answer is option 'D'. Can you explain this answer?
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