The Mystery of Recursive Sequences: Solving f (n + 1) = f(n) – 2. if f(1) = 18, what is f(5)?

Welcome, curious minds, to a journey through the captivating realm of recursive sequences. Today, we embark on a quest to decipher the enigmatic equation ‘f (n + 1) = f(n) – 2. if f(1) = 18, what is f(5)?’ and unveil its secrets, all while unraveling the mystery of what ‘f(5)’ truly represents when ‘f(1) = 18’.

Understanding Recursive Sequences

Before diving headfirst into the depths of our equation, let’s first grasp the concept of recursive sequences, particularly as it pertains to the equation ‘f (n + 1) = f(n) – 2. if f(1) = 18, what is f(5)?’. These sequences are characterized by a rule that defines each term in the sequence based on preceding terms. In simpler terms, each term relies on the one before it, forming a chain of interconnected values.

The Essence of ‘f (n + 1) = f(n) – 2’

Our journey begins with the heart of the matter: the equation ‘f (n + 1) = f(n) – 2. if f(1) = 18, what is f(5)?’. This expression serves as the guiding principle behind our recursive sequence, dictating how each term evolves as we progress along the sequence.

Breaking Down the Equation

To better understand the mechanics at play, let’s break down the components of our equation:

• f (n + 1): This denotes the term that follows a given term, represented by ‘n’. In essence, it’s the next value in our sequence.
• f(n): Here, ‘f(n)’ represents the current term in our sequence.
• – 2: The subtraction of 2 signifies the rule by which each term is derived from its predecessor. In other words, each term is decremented by 2 from the previous one.

Of course! We can find the value of f(5) using the given recursive formula and the fact that f(1) = 18. Here’s how:

Step 1: Understand the formula

The formula f(n + 1) = f(n) – 2 tells us that the value of any term in the sequence (f(n + 1)) is obtained by subtracting 2 from the value of the previous term (f(n)). In other words, each term is 2 less than the one before it.

Step 2: Start with the known value

We know that f(1) = 18. This is our starting point.

Step 3: Apply the formula recursively

To find f(2), we can substitute n = 1 in the formula:

f(2) = f(1) – 2 = 18 – 2 = 16

Now, we know that f(2) = 16.

Step 4: Repeat for f(3), f(4), and f(5)

Following the same logic:

f(3) = f(2) – 2 = 16 – 2 = 14 f(4) = f(3) – 2 = 14 – 2 = 12 f(5) = f(4) – 2 = 12 – 2 = 10

Therefore, f(5) = 10.

Answer:

The value of f(5) is 10.

Navigating the Sequence: From f(1) to f(5)

With a firm grasp of our equation, let’s embark on our journey through the recursive sequence, starting from ‘f(1)’ and advancing towards ‘f(5)’.

Setting the Stage: f(1) = 18

Our expedition begins with the initial value of ‘f(1)’, which is revealed to be 18. This serves as our starting point, the foundation upon which we build the subsequent terms of our sequence.

Unraveling the Sequence: Computing f(2), f(3), and f(4)

As we venture forth, each step brings us closer to unraveling the mystery of ‘f(5)’. Let’s compute the values of ‘f(2)’, ‘f(3)’, and ‘f(4)’ to pave the way for our final destination.

Computing f(2)

Substituting ‘n = 1’ into our equation, we find:

$f(2)=f(1)-2$ $f(2)=18-2$ $f(2)=16$

Thus, ‘f(2)’ is revealed to be 16.

Computing f(3)

Continuing our journey, we calculate ‘f(3)’ by substituting ‘n = 2’:

$f(3)=f(2)-2$ $f(3)=16-2$ $f(3)=14$

Hence, ‘f(3)’ is unveiled as 14.

Computing f(4)

Pressing onward, we ascertain the value of ‘f(4)’ by substituting ‘n = 3’:

$f(4)=f(3)-2$ $f(4)=14-2$ $f(4)=12$

With this, ‘f(4)’ emerges as 12.

The Final Stretch: Determining f(5)

At long last, we arrive at the climax of our journey: the computation of ‘f(5)’. With the values of ‘f(1)’ through ‘f(4)’ in hand, we are poised to unveil the ultimate revelation.

Computing f(5)

Substituting ‘n = 4’ into our equation, we proceed to compute ‘f(5)’:

$f(5)=f(4)-2$ $f(5)=12-2$ $f(5)=10$

And there it is, the culmination of our efforts—’f(5)’ stands revealed as 10.

Conclusion: Unraveling the Enigma

So, to those who pondered the riddle posed by ‘f(1) = 18, what is f(5)?’, the answer resounds clear: ‘f(5) = 10’. With this revelation, we bid adieu to our journey, armed with newfound knowledge and a deeper understanding of recursive sequences.