Geometric Sequence: Recursive Vs. Explicit Formula

Hey math whizzes and sequence lovers! Today, we're diving deep into the cool world of geometric sequences. You know, those awesome patterns where each term is found by multiplying the previous one by a constant number – that number is called the common ratio. We've got Sebastian here, who's given us a sweet recursive formula: $f(x+1)=4f(x)$. This tells us that to get the next term, we multiply the current term by 4. That '4' is our common ratio, folks! He also dropped a hint that the second term of this sequence is 12. Now, the big question is, which explicit formula can we use to model this exact same sequence? Let's break it down, shall we? Understanding the relationship between recursive and explicit formulas is super important for mastering sequences. A recursive formula defines a term based on its preceding term(s), often requiring a starting value. Think of it as a set of instructions to get from one step to the next. An explicit formula, on the other hand, allows you to calculate any term directly, without needing to know the previous terms. This is like having a direct map to any destination, rather than following a step-by-step guide. For geometric sequences, the recursive formula typically looks like $a_n = r imes a_{n-1}$, where 'r' is the common ratio. Sebastian's formula, $f(x+1)=4f(x)$, fits this mold perfectly, with $r=4$. The challenge now is to translate this into an explicit formula, which usually takes the form $a_n = a_1 imes r^{n-1}$ or $f(x) = a imes r^{x-1}$, where '$a$' is the first term and '$r$' is the common ratio. We know $r=4$, but we need to find the first term, $f(1)$, or the initial value '$a$'. Sebastian's clue about the second term being 12 is our golden ticket here. So, let's get our thinking caps on and figure this out together!

Unpacking the Recursive Formula and the Second Term Clue

Alright guys, let's really get into what Sebastian's given us. The recursive formula is $f(x+1)=4f(x)$. This means that any term in the sequence is 4 times the term that came right before it. For example, if we knew $f(1)$, then $f(2)$ would be $4 imes f(1)$, $f(3)$ would be $4 imes f(2)$, and so on. This '4' is our common ratio ($r=4$). Now, Sebastian also told us that the second term, $f(2)$, is equal to 12. This is a crucial piece of information! We can use this to find the very first term of the sequence, $f(1)$. Remember the relationship from the recursive formula: $f(2) = 4 imes f(1)$. We know $f(2)=12$, so we can plug that in: $12 = 4 imes f(1)$. To find $f(1)$, we just need to do a little algebraic magic. Divide both sides by 4: $f(1) = 12 / 4$. And voilà! The first term, $f(1)$, is 3. So now we know two critical things about our geometric sequence: the first term is 3, and the common ratio is 4. This is exactly what we need to build our explicit formula. The explicit formula for a geometric sequence is generally given by $f(x) = a imes r^{x-1}$, where '$a$' is the first term and '$r$' is the common ratio. In our case, $a = f(1) = 3$ and $r=4$. So, substituting these values into the general formula, we get $f(x) = 3 imes 4^{x-1}$. This formula allows us to calculate any term in the sequence directly. For instance, to find the second term using this explicit formula, we'd plug in $x=2$: $f(2) = 3 imes 4^{2-1} = 3 imes 4^1 = 3 imes 4 = 12$. This matches the information Sebastian gave us, which is a fantastic sign! Let's check the third term just for kicks. Using the explicit formula: $f(3) = 3 imes 4^{3-1} = 3 imes 4^2 = 3 imes 16 = 48$. Using the recursive formula, $f(3) = 4 imes f(2) = 4 imes 12 = 48$. It matches! This confirms our explicit formula is spot on. The beauty of the explicit formula is its directness. No need to calculate all the preceding terms. Want the 10th term? Just plug in $x=10$ into $f(x) = 3 imes 4^{x-1}$ and you're golden. This is the power of understanding how these different sequence representations work together. It’s all about translating between different ways of describing the same mathematical pattern.

Evaluating the Explicit Formula Options

Now that we've figured out the explicit formula should be $f(x) = 3 imes 4^{x-1}$, let's look at the options Sebastian provided and see which one matches our findings. We're hunting for the formula that accurately models the sequence where the first term is 3 and the common ratio is 4, and specifically where the second term is 12.

• Option A: $f(x)=12(4)^x$ Let's test this one out. If we plug in $x=1$ (for the first term), we get $f(1) = 12(4)^1 = 12 imes 4 = 48$. This doesn't match our first term of 3, nor does it correctly give us the second term as 12 (if $f(1)=48$, then $f(2)$ should be $4 imes 48 = 192$, not 12). So, option A is a no-go.

• Option B: $f(x)=3(4)^{x-1}$ This looks promising! Let's plug in $x=1$: $f(1) = 3(4)^{1-1} = 3(4)^0 = 3 imes 1 = 3$. Perfect, that's our first term! Now let's check the second term by plugging in $x=2$: $f(2) = 3(4)^{2-1} = 3(4)^1 = 3 imes 4 = 12$. Bingo! This matches the second term given by Sebastian. Let's quickly check the recursive relation: $f(x+1) = 3(4)^{(x+1)-1} = 3(4)^x$. And $4f(x) = 4 imes [3(4)^{x-1}] = 4 imes 3 imes 4^{x-1} = 3 imes 4^1 imes 4^{x-1} = 3 imes 4^{1 + (x-1)} = 3 imes 4^x$. Since $f(x+1) = 3(4)^x$ and $4f(x) = 3(4)^x$, the recursive relation holds. This option seems to be our winner!

• Option C: $f(x)=4(12)^x$ Let's test this one. If $x=1$, $f(1) = 4(12)^1 = 4 imes 12 = 48$. Again, this doesn't give us our first term of 3. The common ratio here seems to be 12, not 4. So, option C is incorrect.

• Option D: $f(x)=4(3)^{x-1}$ Let's check this out. If $x=1$, $f(1) = 4(3)^{1-1} = 4(3)^0 = 4 imes 1 = 4$. This doesn't match our first term of 3. The common ratio here is 3, not 4. So, option D is also incorrect.

The Final Verdict: Which Explicit Formula Wins?

After meticulously examining each option, it's crystal clear, guys. Option B, $f(x)=3(4)^{x-1}$, is the explicit formula that perfectly models the geometric sequence Sebastian described. We started with the recursive formula $f(x+1)=4f(x)$ and the information that the second term, $f(2)$, is 12. From this, we deduced that the common ratio ($r$) is 4 and the first term ($f(1)$) must be 3. The general form of an explicit formula for a geometric sequence is $f(x) = a imes r^{x-1}$, where '$a$' is the first term and '$r$' is the common ratio. Plugging in our values, $a=3$ and $r=4$, we arrived at $f(x) = 3 imes 4^{x-1}$. When we tested this formula against the given options, it was the only one that satisfied all conditions: it produced a first term of 3, a second term of 12, and maintained the recursive relationship $f(x+1)=4f(x)$. The other options either had the wrong first term, the wrong common ratio, or both, failing to represent the sequence accurately. So, the correct answer is indeed Option B. It’s a great example of how you can move between different representations of a sequence, using the clues provided to unlock the right formula. Keep practicing, and you'll be a sequence master in no time!