Round the following factors to estimate the products

Fred completen 10 math problems This is 40 percent of the number of math problems he has to do How many math problems must he do

ZanzabumX [31]

He got 4 out of the 10 right how you do that you would divide the two numbers together

8 0

4 years ago

Read 2 more answers

What is a quick and easy way to remember explicit and recursive formulas?

Oliga [24]

I always found derivation to be helpful in remembering. Since this question is tagged as at the middle school level, I assume you've only learned about arithmetic and geometric sequences.

First, remember what these names mean. An arithmetic sequence is a sequence in which consecutive terms are increased by a fixed amount; in other words, it is an additive sequence. If $a_n$ is the $n$ th term in the sequence, then the next term $a_{n+1}$ is a fixed constant (the common difference $d$ ) added to the previous term. As a recursive formula, that's

$a_{n+1}=a_n+d$

This is the part that's probably easier for you to remember. The explicit formula is easily derived from this definition. Since $a_{n+1}=a_n+d$ , this means that $a_n=a_{n-1}+d$ , so you plug this into the recursive formula and end up with

$a_{n+1}=(a_{n-1}+d)+d=a_{n-1}+2d$

You can continue in this pattern, since every term in the sequence follows this rule:

$a_{n+1}=a_{n-1}+2d$
$a_{n+1}=(a_{n-2}+d)+2d$
$a_{n+1}=a_{n-2}+3d$
$a_{n+1}=(a_{n-3}+d)+3d$
$a_{n+1}=a_{n-3}+4d$

and so on. You start to notice a pattern: the subscript of the earlier term in the sequence (on the right side) and the coefficient of the common difference always add up to $n+1$ . You have, for example, $(n-2)+3=n+1$ in the third equation above.

Continuing this pattern, you can write the formula in terms of a known number in the sequence, typically the first one $a_1$ . In order for the pattern mentioned above to hold, you would end up with

$a_{n+1}=a_1+nd$

or, shifting the index by one so that the formula gives the $n$ th term explicitly,

$a_n=a_1+(n-1)d$

Now, geometric sequences behave similarly, but instead of changing additively, the terms of the sequence are scaled or changed multiplicatively. In other words, there is some fixed common ratio $r$ between terms that scales the next term in the sequence relative to the previous one. As a recursive formula,

$a_{n+1}=ra_n$

Well, since $a_n$ is just the term after $a_{n-1}$ scaled by $r$ , you can write

$a_{n+1}=r(ra_{n-1})=r^2a_{n-1}$

Doing this again and again, you'll see a similar pattern emerge:

$a_{n+1}=r^2a_{n-1}$
$a_{n+1}=r^2(ra_{n-2})$
$a_{n+1}=r^3a_{n-2}$
$a_{n+1}=r^3(ra_{n-3})$
$a_{n+1}=r^4a_{n-3}$

and so on. Notice that the subscript and the exponent of the common ratio both add up to $n+1$ . For instance, in the third equation, $3+(n-2)=n+1$ . Extrapolating from this, you can write the explicit rule in terms of the first number in the sequence:

$a_{n+1}=r^na_1$

or, to give the formula for $a_n$ explicitly,

$a_n=r^{n-1}a_1$

6 0

4 years ago

A parallelogram is translated 2 units to the right and 1 unit down and then rotated 90 degrees clockwise about the orgin. Does i

vazorg [7]

Yes because it is a square and if you rotate a square 90 degrees it will stay the same shape

Have a good day!

6 0

3 years ago

In a right triangle, a and b are the lengths of the legs and c is the length of the hypotenuse. If a=9 feet and c=10 feet, what

olchik [2.2K]

Answer:

b = 4.4 feet

Step-by-step explanation:

To find the value of "b", all you need to do is plug the values for "a" and "c" into the pythagorean Theorem and isolate "b".

a² + b² = c² <------ Phythagorean Theorem

9² + b² = 10² <------- Plug in a = 9 and c = 10

81 + b² = 100 <-------- Square both values

b² = 19 <-------- Subtract 81 from both sides

$\sqrt{b^{2} } = \sqrt{19}$ <-------- Square root both sides to remove exponent

b = 4.358...

b = 4.4 feet

5 0

2 years ago

If y varies directly as x and y = 42 when x = 6, find the equation that gives the relationship between x and y.

Troyanec [42]

Answer:

B

Step-by-step explanation:

since K is constant ( the same for every point) we can find k when given any point by dividing the y-coordinate by the x-coordinate.

6 0

3 years ago