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If the sum of n terms of a G.P. with last term 128 and common ratio 2 is 255, 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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Most Upvoted Answer
If the sum of n terms of a G.P. with last term 128 and common ratio 2 ...
Given:
Last term of G.P. = 128
Common ratio = 2
Sum of n terms = 255
To find: Value of n
Formula used: Sum of n terms of a G.P. with first term 'a' and common ratio 'r' is given by
S_n = a(1 - r^n)/(1 - r)
Calculation:
Let the first term of the G.P be a.
Given, last term of the G.P. = 128
Using formula for last term of G.P., we get
128 = a(2^(n-1))
Given, sum of n terms of the G.P. = 255
Using formula for sum of n terms of G.P., we get
255 = a(1 - 2^n)/(1 - 2)
Simplifying, we get
255 = a(2^n - 1)
Dividing the above two equations, we get
128/255 = 2^(n-1)/(2^n - 1)
Simplifying, we get
2^(n-1) = 1/2
Taking log on both sides, we get
(n-1) log2 = log(1/2)
Solving for n, we get
n = 8
Therefore, the value of n is 8.
Hence, the correct option is (a) 8.
Last term of G.P. = 128
Common ratio = 2
Sum of n terms = 255
To find: Value of n
Formula used: Sum of n terms of a G.P. with first term 'a' and common ratio 'r' is given by
S_n = a(1 - r^n)/(1 - r)
Calculation:
Let the first term of the G.P be a.
Given, last term of the G.P. = 128
Using formula for last term of G.P., we get
128 = a(2^(n-1))
Given, sum of n terms of the G.P. = 255
Using formula for sum of n terms of G.P., we get
255 = a(1 - 2^n)/(1 - 2)
Simplifying, we get
255 = a(2^n - 1)
Dividing the above two equations, we get
128/255 = 2^(n-1)/(2^n - 1)
Simplifying, we get
2^(n-1) = 1/2
Taking log on both sides, we get
(n-1) log2 = log(1/2)
Solving for n, we get
n = 8
Therefore, the value of n is 8.
Hence, the correct option is (a) 8.
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