How many terms are there in the G.P. 3, 6, 12, 24, ... , 384 ?

To find the number of terms in a geometric progression (G.P.), we can use the formula:

n = [log(last term/first term) / log(common ratio)] + 1

In this G.P., the first term (a) is 3, the common ratio (r) is 2 (6/3 = 2, 12/6 = 2, etc.), and the last term is 384.

So, substituting these values into the formula, we get:

n = [log(384/3) / log(2)] + 1

Calculating the values inside the brackets first:

log(384/3) = log(128) = 2.10720997 (approx)

log(2) = 0.301029995 (approx)

So, n = [2.10720997 / 0.301029995] + 1 = 7 + 1 = 8

So, there are 8 terms in the G.P. 3, 6, 12, 24, ... , 384.