Question

Consider the following function: int secret(int m, int n) { int temp = 0; for (int i = 1; i < abs(n); i++) temp = temp + i * m; return temp; } a. What is the output of the following C++ statements? i. cout << secret(3, 6) << endl; ii. cout << secret(5, -4) << endl; b. What does the function secret do?

Short answer

i. Output is 45. ii. Output is 30. The function calculates the sum of multiples of $m$ from 1 to $|n|-1$.

Step-by-step solution

Analyze the Function

We begin by analyzing the function 'secret'. It takes two integer inputs, $m$ and $n$. It initializes a variable `temp` to 0. The function then runs a loop from 1 to $|n|-1$ (i.e., the absolute value of $n$ minus 1). In each iteration, it adds the product of $i$ and $m$ to `temp`. The function then returns `temp`.

Calculate secret(3, 6)

For $m=3$ and $n=6$, the loop runs from 1 to 5. 1. When $i=1$, `temp` becomes $0+1\times 3=3$. 2. When $i=2$, `temp` becomes $3+2\times 3=9$. 3. When $i=3$, `temp` becomes $9+3\times 3=18$. 4. When $i=4$, `temp` becomes $18+4\times 3=30$. 5. When $i=5$, `temp` becomes $30+5\times 3=45$.The final value of `temp` is 45.

Calculate secret(5, -4)

For $m=5$ and $n=-4$, the loop runs from 1 to 3 (since $|n|=4$). 1. When $i=1$, `temp` becomes $0+1\times 5=5$. 2. When $i=2$, `temp` becomes $5+2\times 5=15$. 3. When $i=3$, `temp` becomes $15+3\times 5=30$.The final value of `temp` is 30.

Determine the Function Purpose

The function calculates a sum of products where each product is the loop index $i$ multiplied by $m$, for values $i$ ranging from 1 to $|n|-1$. Essentially, it's calculating the sum of an arithmetic sequence where the common difference equals $m$.

Key concepts

Loops in C++

Loops in C++ are fundamental constructs that allow you to execute a block of code repeatedly. In the context of the C++ function provided, a `for` loop is used to iterate a specific number of times.
The syntax of the `for` loop includes three parts: initialization, condition, and increment. This loop runs from an initial integer value to a condition-defined limit. Each iteration modifies the loop variable.

• Initialization: Sets the starting point of the loop. Here, it's `int i = 1`, which means the loop starts with `i` equal to 1.
• Condition: Dictates when the loop stops. In this function, it continues as long as `i` is less than `|n|` (the absolute value of `n`).
• Increment: Updates the loop variable after each iteration. Here, `i` increases by 1 each time (`i++`).

Loops are incredibly useful in programming, allowing us to automate tasks, repeat calculations, or manipulate data structures without duplicating code. By understanding how loops work, you can efficiently control and manage script behavior.

Arithmetic Sequences

An arithmetic sequence is a series of numbers where each term increases by a constant value, called the "common difference." In programming, they're useful for calculating sums efficiently.
In the provided function, each term in the sequence is multiplied by the variable `m`, with `i` representing each successive term in the iteration.
Here’s how it applies:

• First term: In each case, the first term is `1 * m`.
• Subsequent terms: Continue adding `m` multiplied by the current index `i` to the cumulative sum.
• Number of terms: This is determined by ` |n| - 1`, the extent to which the loop iterates.

These steps accumulate to create the total sum of the sequence up to the term before the absolute value of `n`. This structure helps solve problems efficiently without calculating each step individually manually.

Absolute Value Function

The absolute value function, denoted `abs()` in C++, returns the non-negative value of a number. It's crucial when you need to ensure computations consider the magnitude rather than the sign of a number.
In the `secret` function, the absolute value of `n` controls the number of loop iterations. For example, if `n` is -4, `abs(n)` becomes 4, which dictates how many times the loop will run.

• Purpose: Eliminates the effect of the sign, focusing purely on size or magnitude, which ensures consistent behavior regardless of `n`'s sign.
• Usage: Particularly useful in mathematical problems, distance calculations, or wherever symmetry around zero is needed.

The function's reliance on `abs(n)` exemplifies its role in controlling logic flow, facilitating versatile and predictable program execution.