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If the sum of n terms of a G.P. with first term 1 and common ratio 1/2 is 1+127/128, the value of n is ___________.
- a)8
- b)5
- c)3
- d)None
Correct answer is option 'A'. Can you explain this answer?
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If the sum of n terms of a G.P. with first term 1 and common ratio 1/2...
Given information:
- First term of the G.P. is 1.
- Common ratio is 1/2.
- Sum of n terms of the G.P. is 1 127/128.
To find: The value of n.
Solution:
Let's use the formula for the sum of n terms of a G.P. with first term 'a' and common ratio 'r':
S_n = a(1 - r^n)/(1 - r)
Substituting the given values, we get:
1 127/128 = 1(1 - (1/2)^n)/(1 - 1/2)
1 127/128 = 2(1 - (1/2)^n)
1 - (1/2)^n = (1 127/128)/2
1 - (1/2)^n = 63/128
Now, we can solve for n using logarithms:
log(1 - (1/2)^n) = log(63/128)
1/2^n = 63/128
n*log(1/2) = log(63/128)
n = log(63/128)/log(1/2)
n = 8
Therefore, the value of n is 8, which is option (a).
- First term of the G.P. is 1.
- Common ratio is 1/2.
- Sum of n terms of the G.P. is 1 127/128.
To find: The value of n.
Solution:
Let's use the formula for the sum of n terms of a G.P. with first term 'a' and common ratio 'r':
S_n = a(1 - r^n)/(1 - r)
Substituting the given values, we get:
1 127/128 = 1(1 - (1/2)^n)/(1 - 1/2)
1 127/128 = 2(1 - (1/2)^n)
1 - (1/2)^n = (1 127/128)/2
1 - (1/2)^n = 63/128
Now, we can solve for n using logarithms:
log(1 - (1/2)^n) = log(63/128)
1/2^n = 63/128
n*log(1/2) = log(63/128)
n = log(63/128)/log(1/2)
n = 8
Therefore, the value of n is 8, which is option (a).
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If the sum of n terms of a G.P. with first term 1 and common ratio 1/2 is 1+127/128, the value of n is ___________.a)8b)5c)3d)NoneCorrect answer is option 'A'. Can you explain this answer?
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