How many terms are there in the geometric progression 3, 6, 12, 24, ..., 384?

Introduction / Context:
This problem involves a geometric progression (G.P.), a sequence in which each term after the first is obtained by multiplying the previous term by a constant ratio. We are given the first few terms and the last term and asked how many terms are in the sequence. This tests understanding of the formula for the n-th term of a geometric progression and basic logarithmic or exponential reasoning.

Given Data / Assumptions:

- The G.P. starts with 3, followed by 6, 12, 24 and continues up to 384. - The first term a is 3. - The common ratio r is 6 ÷ 3 = 2. - The last term given is 384 and is part of the sequence.

Concept / Approach:
In a geometric progression, the n-th term T_n is given by T_n = a * r^(n − 1), where a is the first term and r is the common ratio. Here, we know a, r and T_n, and we need to find n. So we solve 3 * 2^(n − 1) = 384 for n. This involves dividing by 3 and then identifying a power of 2 that equals the result.

Step-by-Step Solution:
Step 1: Note that a = 3 and r = 2, because 6 ÷ 3 = 2, 12 ÷ 6 = 2 and 24 ÷ 12 = 2. Step 2: Let the number of terms be n. Then the n-th term is T_n = 3 * 2^(n − 1). Step 3: Given that T_n = 384, set up the equation 3 * 2^(n − 1) = 384. Step 4: Divide both sides by 3 to isolate the power of 2: 2^(n − 1) = 384 ÷ 3 = 128. Step 5: Recognize that 128 is a power of 2. Specifically, 2^7 = 128. Step 6: Therefore, 2^(n − 1) = 2^7, which implies n − 1 = 7. Step 7: Solve for n to get n = 8.

Verification / Alternative check:
We can list the terms: 3 (first), 6 (second), 12 (third), 24 (fourth), 48 (fifth), 96 (sixth), 192 (seventh) and 384 (eighth). Counting them confirms that 384 is the eighth term in the sequence, so there are eight terms in total. This matches the result from the formula.

Why Other Options Are Wrong:
Options B, C and D suggest 9, 10 or 11 terms. If there were 9 terms, then T_9 would be 3 * 2^8 = 3 * 256 = 768, which is larger than 384. Hence 384 would not be the last term. Similarly, for 10 or 11 terms, the last terms would be 1536 or 3072, which do not match the given last term.

Common Pitfalls:
One common mistake is to miscount the terms when listing the progression, accidentally including or skipping a term. Another is to confuse the formula for the n-th term with the formula for the sum of terms. Remember that T_n = a * r^(n − 1) is the correct formula for the n-th term in a geometric sequence.

Final Answer:
The geometric progression 3, 6, 12, 24, ..., 384 contains 8 terms, which corresponds to option A.