how many terns has the g.p whose second term is half and the common ratio and the last term are 1 over 4 and 1 over one hundred and twenty eight
Fix your post, it makes no sense
To find the number of terms in a geometric progression (G.P.), we can use the formula:
n = (log(L / A)) / log(r)
where:
n = number of terms
L = last term
A = first term
r = common ratio
In this case, we are given that the second term is half, the common ratio is 1/4, and the last term is 1/128.
Let's begin by finding the first term, A. Since the second term is half, we can write it as:
A * r = 1/2
From this equation, we can solve for A:
A = (1/2) / r
Now, let's substitute the values into the formula to find the number of terms:
n = (log(L / A)) / log(r)
n = (log(1/128 / [(1/2) / r])) / log(1/4)
n = (log(1/128) + log(r / (1/2))) / log(1/4)
n = (log(1/128) + log(2r)) / log(1/4)
Since log(1/4) is -2 (logarithm base 10), we can simplify further:
n = (log(1/128) + log(2r)) / (-2)
Now, let's evaluate the logarithms:
n = (log(1) - log(128) + log(2r)) / (-2)
n = (0 - log(128) + log(2r)) / (-2)
Using log properties, log(128) can be expressed as log(2^7):
n = (- log(2^7) + log(2r)) / (-2)
n = (-7 * log(2) + log(2r)) / (-2)
Finally, we can simplify the expression:
n = (7 * log(2) - log(2r)) / 2
So, the number of terms in the geometric progression is (7 * log(2) - log(2r)) / 2. Evaluating this expression will give you the answer.