(x^2 - 10x + 2) ÷ (x

Find the sum of the positive integers less than 200 which are not multiples of 4 and 7

taurus [48]

Answer:

$12942$ is the sum of positive integers between $1$ (inclusive) and $199$ (inclusive) that are not multiples of $4$ and not multiples $7$ .

Step-by-step explanation:

For an arithmetic series with:

$a_1$ as the first term,
$a_n$ as the last term, and
$d$ as the common difference,

there would be $\displaystyle \left(\frac{a_n - a_1}{d} + 1\right)$ terms, where as the sum would be $\displaystyle \frac{1}{2}\, \displaystyle \underbrace{\left(\frac{a_n - a_1}{d} + 1\right)}_\text{number of terms}\, (a_1 + a_n)$ .

Positive integers between $1$ (inclusive) and $199$ (inclusive) include:

$1,\, 2,\, \dots,\, 199$ .

The common difference of this arithmetic series is $1$ . There would be $(199 - 1) + 1 = 199$ terms. The sum of these integers would thus be:

$\begin{aligned}\frac{1}{2}\times ((199 - 1) + 1) \times (1 + 199) = 19900 \end{aligned}$ .

Similarly, positive integers between $1$ (inclusive) and $199$ (inclusive) that are multiples of $4$ include:

$4,\, 8,\, \dots,\, 196$ .

The common difference of this arithmetic series is $4$ . There would be $(196 - 4) / 4 + 1 = 49$ terms. The sum of these integers would thus be:

$\begin{aligned}\frac{1}{2}\times 49 \times (4 + 196) = 4900 \end{aligned}$

Positive integers between $1$ (inclusive) and $199$ (inclusive) that are multiples of $7$ include:

$7,\, 14,\, \dots,\, 196$ .

The common difference of this arithmetic series is $7$ . There would be $(196 - 7) / 7 + 1 = 28$ terms. The sum of these integers would thus be:

$\begin{aligned}\frac{1}{2}\times 28 \times (7 + 196) = 2842 \end{aligned}$

Positive integers between $1$ (inclusive) and $199$ (inclusive) that are multiples of $28$ (integers that are both multiples of $4$ and multiples of $7$ ) include:

$28,\, 56,\, \dots,\, 196$ .

The common difference of this arithmetic series is $28$ . There would be $(196 - 28) / 28 + 1 = 7$ terms. The sum of these integers would thus be:

$\begin{aligned}\frac{1}{2}\times 7 \times (28 + 196) = 784 \end{aligned}$ .

The requested sum will be equal to:

the sum of all integers from $1$ to $199$ ,
minus the sum of all integer multiples of $4$ between $1\!$ and $199\!$ , and the sum integer multiples of $7$ between $1$ and $199$ ,
plus the sum of all integer multiples of $28$ between $1$ and $199$ - these numbers were subtracted twice in the previous step and should be added back to the sum once.

That is:

$19900 - 4900 - 2842 + 784 = 12942$ .

8 0

4 years ago

You charge customers $18.75 for lube service, parts, and labor. Assuming you take one hour's labor at $15, and the parts cost $3

hram777 [196]

Let
T---------> <span>the tax rate on the parts in $

we know that
$18.75=$15+$3.50+$T
T=18.75-15-3.5
T=$0.25
so
$3.50----------> 100%
$0.25--------> x
x=0.25*100/3.50------> x=7.14%------> x=7.1%

the answer is
7.1%</span>

5 0

3 years ago

PLZ HELP WILL GIVE BRAINLST

MArishka [77]

B is plus square so you would add

7 0

4 years ago

A car traveling at a certain speed will travel 76 feet per second. How man yards will the car travel in 120 seconds if it ma bra

lys-0071 [83]

The equation is 76*120/3.

76*120/3
9,120/3
3,040

The car will travel 3,040 yards in 120 seconds at that speed.

7 0

4 years ago

Whats the length of the line segment EF

slamgirl [31]

Answer:

The length is 14 units

Step-by-step explanation:

Take the value at point F and subtract the value at point E

F = 19

E = 5

19 -5 =14

The length is 14 units

7 0

3 years ago

(x^2 - 10x + 2) ÷ (x - 2)​

(x^2 - 10x + 2) ÷ (x - 2)