The question asks whether -1/127 is any term of the series 4,-2,1,. of G.P. In order to answer this question, we need to first understand what a G.P. is and how to find its terms.


A G.P. or geometric progression is a series of numbers where each term is found by multiplying the previous term by a constant ratio. The formula for finding the nth term of a G.P. is given by: a_n = a_1 * r^(n-1), where a_1 is the first term, r is the common ratio, and n is the term number.


Using the formula for finding the nth term of a G.P., we can find the terms of the series 4,-2,1,. The first term is 4 and the common ratio can be found by dividing any term by the previous term. In this case, we can divide -2 by 4 to get -1/2 as the common ratio.

Therefore, the nth term of the series is given by: a_n = 4 * (-1/2)^(n-1)


Now that we have the formula for finding the nth term of the series, we can check whether -1/127 is a term. We can do this by solving the formula for n and seeing if we get a whole number.

-1/127 = 4 * (-1/2)^(n-1)

Dividing both sides by 4, we get:

-1/508 = (-1/2)^(n-1)

Taking the logarithm of both sides, we get:

log(-1/508) = (n-1) * log(-1/2)

Solving for n, we get:

n = 1 - (log(-1/508) / log(-1/2))

However, we cannot take the logarithm of a negative number, so -1/127 is not a term of the series 4,-2,1,. of G.P.


In conclusion, -1/127 is not a term of the series 4,-2,1,. of G.P. We were able to determine this by finding the formula for the nth term of the series and solving for n using the given value of -1/127.