B5 - Analysis of Calculator Programs 1. The Alpha performer has two commands assigned to them: 1. add 1 2. multiply

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natural number; b ≥ 2). By executing the first command, Alpha increases the number on the screen by 1, and by executing the second command, multiplies that number by b. We are given the program 11221, which converts the number 2 into the number 197.

To determine the value of b, let"s step through the program and analyze its effect on the number. Starting with the number 2, we follow each command:

1. The command "add 1" increases the number by 1. So, after executing this command, the number becomes 2 + 1 = 3.
2. The command "multiply by b" multiplies the number by b. At this point, the number is 3. Therefore, after this command, the number becomes 3b.
3. The command "add 1" increases the number by 1. So, the number becomes 3b + 1.
4. The command "add 1" increases the number by 1 again. The number becomes 3b + 2.
5. Finally, the command "multiply by b" multiplies the number by b. After this command, the number becomes (3b + 2) * b.

According to the given information, the program converts the number 2 into the number 197. Therefore, we can set up the equation:

\[(3b + 2) * b = 197\]

Expanding the equation, we get:

\[3b^2 + 2b = 197\]

To solve this equation, we need to find the value of b that satisfies it. We can rearrange it into a quadratic equation in standard form:

\[3b^2 + 2b - 197 = 0\]

Now we can solve this quadratic equation using various methods such as factoring, completing the square, or using the quadratic formula. Let"s use the quadratic formula to find the roots of this equation:

The quadratic formula states that for an equation of the form \(ax^2 + bx + c = 0\), the solutions for \(x\) are given by:

\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]

In our case, the equation is \(3b^2 + 2b - 197 = 0\), so \(a = 3\), \(b = 2\), and \(c = -197\). Substituting these values into the quadratic formula, we get:

\[b = \frac{{-2 \pm \sqrt{{2^2 - 4 \cdot 3 \cdot (-197)}}}}{{2 \cdot 3}}\]

Simplifying inside the square root, we have:

\[b = \frac{{-2 \pm \sqrt{{4 + 2364}}}}{{6}}\]

Further simplifying:

\[b = \frac{{-2 \pm \sqrt{{2368}}}}{{6}}\]

We can calculate the value inside the square root:

\[b = \frac{{-2 \pm \sqrt{{8 \cdot 296}}}}{{6}}\]

\[b = \frac{{-2 \pm \sqrt{{8}} \cdot \sqrt{{296}}}}{{6}}\]

\[b = \frac{{-2 \pm 2 \sqrt{{74}}}}{{6}}\]

Simplifying the expression:

\[b = \frac{{-1 \pm \sqrt{{74}}}}{{3}}\]

Since b represents a natural number and is greater than or equal to 2, the only valid solution is:

\[b = \frac{{-1 + \sqrt{{74}}}}{{3}}\]

Therefore, the value of b that satisfies the given conditions is approximately equal to 1.867.

Answer: Значение b, которое удовлетворяет условию, равно приблизительно 1.867.