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# What is the Meaning of the Mathematical Expression “if f(1) = 160 And f(n + 1) = –2f(n)”?

The mathematical expression “if f(1) = 160 And f(n + 1) = –2f(n)” is used to represent a recursive function that increases or decreases by two times the previous value. This expression can be used to determine the value of an unknown function at any given point.

In this section, we will explore how to use this expression to solve for the value of f(4).

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## If f(1) = 160 And f(n + 1) = –2f(n), What Is f(4)?

The mathematical expression “if f(1) = 160 and f(n+1) = -2f(n)” is a recursive formula that describes a sequence of numbers. In this formula, f(1) = 160 means that the first term in the sequence is 160. The subsequent terms are found by multiplying the previous term by -2.

For example, the second term would be f(2) = -2*f(1) = -2*160 = -320. The third term would be f(3) = -2*f(2) = -2*(-320) = 640. And so on.

Therefore, “f(1) = 160” is the starting point for the sequence, and it informs us that the first term in the sequence has a value of 160. From there, we can use the recursive formula to calculate the rest of the terms in the sequence.

Pro tip: Recursive formulas can be useful in a wide range of mathematical problems, from sequences to calculus to computer science.

### Explaining the meaning of f(n + 1) = –2f(n) and its relation to f(1) = 160

The mathematical expression “if f(1) = 160 and f(n + 1) = -2f(n)” represents a recursive sequence where the value of each term depends on the previous term. In this case, the value of f(n+1) is equal to negative twice the value of f(n).

To find the value of any term in the sequence, you need to plug in the value of n into the equation and work backwards from f(1). For example, to find f(2), you would substitute n=1 into f(n+1)=-2f(n) to get f(2)=-2*f(1)=-2*160=-320.

This sequence demonstrates exponential decay, which means that each term in the sequence is a fraction of the previous term. In this case, the value of each term is half of the previous term.

Therefore, with a starting value of f(1)=160, the sequence quickly approaches zero.

### Deriving the values of f(2), f(3), and f(4) using f(n + 1) = –2f(n) and f(1) = 160

The mathematical expression “if f(1) = 160 and f(n + 1) = -2f(n)” is a recursive formula used to derive the values of f(2), f(3), and f(4).

Here are the steps to follow:

Plug in the value of n into the formula to find the consecutive terms of the sequence:

 f(2) = -2f(1) = -2(160) = -320 f(3) = -2f(2) = -2(-320) = 640 f(4) = -2f(3) = -2(640) = -1280

Therefore, the values of f(2), f(3), and f(4) are -320, 640, and -1280, respectively.

## Finding the Value of f(4) Using the Mathematical Expression

In order to find the value of f(4), we will need to break down the given mathematical expression and calculate the value of f(4) based on the given equation. The mathematical expression given states that if f(1) = 160 and f(n + 1) = –2f(n), then we can solve f(4) by using a recursive approach. Let’s take a look at the calculation process.

### Plugging in the value of n as 3 to find f(4)

The mathematical expression “if f(1) = 160 and f(n + 1) = -2f(n)” is a recursive formula that defines a sequence of values for the function f(x). In this sequence, the initial value of the function is 160 when x=1, and every subsequent value is obtained by multiplying the previous value by -2.

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To find the value of f(4), we can apply the recursive formula using the given initial value of f(1) = 160:

 f(2) = -2f(1) = -2(160) = -320 f(3) = -2f(2) = -2(-320) = 640 f(4) = -2f(3) = -2(640) = -1280

Thus, plugging in the value of n as 3 in the recursive formula gives us f(4) = -1280.

### Showing the step-by-step calculation to find f(4)

The mathematical expression “if f(1) = 160 and f(n + 1) = –2f(n)” is a recursive function that defines the value of f(n) in terms of the value of f(n-1). Here’s how you can find the value of f(4) using this expression:

 f(2) = -2f(1) = -2(160) = -320 f(3) = -2f(2) = -2(-320) = 640 f(4) = -2f(3) = -2(640) = -1280

Therefore, f(4) = -1280. This means that if you plug in 4 for n in the given expression, the value of the function is -1280. Recursive functions like these are used in various areas of mathematics and computer science to define sequences of numbers or solve complex problems.

Pro tip: Simplify your calculation by working methodically and keeping track of your steps carefully.

### Presenting the final answer for f(4) using the given mathematical expression

The mathematical expression “if f(1) = 160 And f(n + 1) = –2f(n)” can be used to find the value of f(4) by following a simple procedure.

 First, solve for f(2) using the given expression: f(2) = -2 * f(1) = -2 * 160 = -320. Next, solve for f(3) using the same expression: f(3) = -2 * f(2) = -2 * (-320) = 640. Finally, solve for f(4) using the expression once again: f(4) = -2 * f(3) = -2 * 640 = -1280.

Therefore, the final answer for f(4) using the given mathematical expression is -1280.

Pro tip: It is always advisable to double-check your calculations to avoid errors while solving mathematical expressions.