What's the intuition behind the equation 1+2+3+⋯=−1121+2+3+⋯=−112 ?

Answer:

Therefore, the correct option is: (C.) [tex]\( f(x) = 2 \cdot 4^x \)[/tex]

Explanation:

To determine the exponential function that passes through the given points, we can use the general form of an exponential function:

[tex]\[ f(x) = a \cdot b^x \][/tex]

Where:

[tex]\( a \)[/tex] is the initial value or the value of the function when [tex]\( x = 0 \).[/tex]

[tex]\( b \)[/tex] is the base of the exponential function.

Given the points [tex]\((1, 8)\) and \((4, 512)\)[/tex], we can substitute these coordinates into the general form and solve for [tex]\( a \) and \( b \).[/tex]

Using the point [tex]\((1, 8)\):[/tex]

[tex]\[ 8 = a \cdot b^1 \][/tex]

[tex]\[ 8 = a \cdot b \][/tex]

Using the point [tex]\((4, 512)\):[/tex]

[tex]\[ 512 = a \cdot b^4 \][/tex]

We can divide the equation obtained from the second point by the equation obtained from the first point to eliminate [tex]\( a \)[/tex] and solve for [tex]\( b \):[/tex]

[tex]\[ \frac{512}{8} = \frac{a \cdot b^4}{a \cdot b} \][/tex]

[tex]\[ 64 = b^3 \][/tex]

Taking the cube root of both sides:

[tex]\[ b = 4 \][/tex]

Now that we have found the value of [tex]\( b \)[/tex], we can substitute it back into one of the equations to find \( a \). Let's use the first point [tex]\((1, 8)\):[/tex]

[tex]\[ 8 = a \cdot 4 \][/tex]

[tex]\[ a = 2 \][/tex]

So, the exponential function that passes through the points [tex]\((1, 8)\) and \((4, 512)\)[/tex] is:

[tex]\[ f(x) = 2 \cdot 4^x \][/tex]

There are 1000 three digit locker combinations are possible if the digits are able to repeat.

The correct option is B.

If the digits are able to repeat, there are 10 options for each digit (0-9) in a three-digit combination.

Since each digit can be chosen independently, the total number of different three-digit combinations is given by:

Number of combinations

= Number of options for the first digit x Number of options for the second digit x Number of options for the third digit

Number of combinations = 10 x 10 x 10 = 1000

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Final answer:

To solve an equation by factoring, find the factors, set them equal to zero, and solve for the variable.

Explanation:

To solve an equation by factoring, you need to find the factors of the equation and set each factor equal to zero. Then, solve each equation separately to find the values of the unknown variable. Let's take an example:

If we have the equation [tex]x^2 - 5x + 6 = 0[/tex]:

This method of factoring and setting each factor equal to zero is a systematic approach to solving quadratic equations, providing a structured way to identify the solutions for the unknown variable. The application of this technique is fundamental in algebra, enabling the efficient determination of roots in quadratic expressions and contributing to a comprehensive understanding of solving equations.

4.2 Adding up the two equivalent fractions 

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

(x+5) - (10 * 7) / 7 = x - 65 / 7

Equation at the end of step 4:

x - 65 / 7 > 0

Step 5: 5.1 Multiply both sides by 7

All equation has always even numbers of imaginary roots because they always come in pairs. Therefore there must be either 2 imaginary roots, or zero. 

To determine the types of roots, we could try synthetic division by some of the factors of the number -32. The simplest factors are +1 or -1. Using this, we can quickly show that (x + 1) is a factor, and leaving (x² + 4x - 32) as the remainder. This remainder equation then factorises to (x - 4) (x + 8). 

So to conclude there are three real zeros which are:

x = -1

x = 4

x = -8

Answer:

never

Step-by-step explanation:

In a 30 - 60 - 90 degree triangle , the common ratio is

1  : [tex]\sqrt{3}[/tex] : 2

Shorter leg that is opposite to 30 degree = 1

Longer leg that is opposite to 60 degree = [tex]\sqrt{3}[/tex]

Hypotenuse is opposite to 90 degree = 2

Shorter leg is half the length of the hypotenuse as per the ratio

Long leg is never the half of the hypotenuse

Answer:

y = 3/7 x - 12/7

Step-by-step explanation:

The slope is (0+3)/(4+3) = 3/7

So, now you have a point and a slope, so use the point-slope form. That gives you

y-0 = 3/7 (x-4)

Now just rearrange to slope-intercept form.

y = 3/7 x - 12/7

Answer:

The answer is option D.

Step-by-step explanation:

A conditional statement is a statement where we can symbolize the given statements in a p and q parts. And this contains if-then clause. It says, if p, then q. Here p is a hypothesis and q is a conclusion. Inverse of a statement means the negative form of the statement in the same order. If not p, then not q.

Given statement is :

If a polygon is regular, then it has congruent angles and congruent sides.

The inverse is :

If a polygon is not regular, then it does not have congruent angles and congruent sides.

Answer:

250%

Step-by-step explanation:

The value of percentage of a number 2.5 is,

⇒ 250%

We have to given that,

To change a number 2.5 into percent.

Since, A number or ratio that can be expressed as a fraction of 100 or a relative value indicating hundredth part of any quantity is called percentage.

Now, We can change it as,

⇒ 2.5 × 100%

⇒ 250%

Therefore, The value of percentage of a number 2.5 is,

⇒ 250%

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