Divide. 6570 ÷ 16 A. 410 B. 410 1/16 C. 410 6/16 D. 410 10/16

Let a1, a2, a3, ... be a sequence of positive integers in arithmetic progression with common difference

Bezzdna [24]

Since $a_1,a_2,a_3,\cdots$ are in arithmetic progression,

$a_2 = a_1 + 2$

$a_3 = a_2 + 2 = a_1 + 2\cdot2$

$a_4 = a_3+2 = a_1+3\cdot2$

$\cdots \implies a_n = a_1 + 2(n-1)$

and since $b_1,b_2,b_3,\cdots$ are in geometric progression,

$b_2 = 2b_1$

$b_3=2b_2 = 2^2 b_1$

$b_4=2b_3=2^3b_1$

$\cdots\implies b_n=2^{n-1}b_1$

Recall that

$\displaystyle \sum_{k=1}^n 1 = \underbrace{1+1+1+\cdots+1}_{n\,\rm times} = n$

$\displaystyle \sum_{k=1}^n k = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}2$

It follows that

$a_1 + a_2 + \cdots + a_n = \displaystyle \sum_{k=1}^n (a_1 + 2(k-1)) \\\\ ~~~~~~~~ = a_1 \sum_{k=1}^n 1 + 2 \sum_{k=1}^n (k-1) \\\\ ~~~~~~~~ = a_1 n + n(n-1)$

so the left side is

$2(a_1+a_2+\cdots+a_n) = 2c n + 2n(n-1) = 2n^2 + 2(c-1)n$

Also recall that

$\displaystyle \sum_{k=1}^n ar^{k-1} = \frac{a(1-r^n)}{1-r}$

so that the right side is

$b_1 + b_2 + \cdots + b_n = \displaystyle \sum_{k=1}^n 2^{k-1}b_1 = c(2^n-1)$

Solve for $c$ .

$2n^2 + 2(c-1)n = c(2^n-1) \implies c = \dfrac{2n^2 - 2n}{2^n - 2n - 1} = \dfrac{2n(n-1)}{2^n - 2n - 1}$

Now, the numerator increases more slowly than the denominator, since

$\dfrac{d}{dn}(2n(n-1)) = 4n - 2$

$\dfrac{d}{dn} (2^n-2n-1) = \ln(2)\cdot2^n - 2$

and for $n\ge5$ ,

$2^n > \dfrac4{\ln(2)} n \implies \ln(2)\cdot2^n - 2 > 4n - 2$

This means we only need to check if the claim is true for any $n\in\{1,2,3,4\}$ .

$n=1$ doesn't work, since that makes $c=0$ .

If $n=2$ , then

$c = \dfrac{4}{2^2 - 4 - 1} = \dfrac4{-1} = -4 < 0$

If $n=3$ , then

$c = \dfrac{12}{2^3 - 6 - 1} = 12$

If $n=4$ , then

$c = \dfrac{24}{2^4 - 8 - 1} = \dfrac{24}7 \not\in\Bbb N$

There is only one value for which the claim is true, $c=12$ .

3 0

1 year ago

casey ran a 75 meter race in 10 seconds her distances at certain times during the race are shown in the table . what is the doma

Troyanec [42]

The domain of the graph is everything x is allowed to equal. Parentheses are used to show that x can be everything up to, but not including that number. Brackets are used to show x can be anything up to, and including, that number. The comma between the two numbers indicates that x can be anything between those two numbers.

Here, the time goes on the x axis and the distance will go on the y axis.
Time: 0 1 2 3 4 5 6 7 8 9 10
Distance: 0.0 5.5 12.0 20.0 28.5 36.0 44.0 56.5 64.0 75.0

Since we are dealing with the domain, we really only need to focus on the time.
Time: 0 1 2 3 4 5 6 7 8 9 10

From this, we see that the values go from 0 to 10. This means that x can be anything from 0 to 10. Because there are y values that correlate with the numbers 0 and 10, we know that the domain will include 0 and 10. This indicates we must use brackets.

Knowing all of this information, we can now write our domain as:
[0, 10]

7 0

3 years ago

Aidan took a 1020 km trip, travelling by Bus and By Train. The bus travelled at 105 km/h, and the train travelled at 160 km/h. T

bazaltina [42]

Answer:

523.6 km

Step-by-step explanation:

Let x represent the distance traveled by train. Then 1020-x is the distance traveled by bus. The travel times are given by the relation ...

time = distance/speed

Then the total travel time in hours is ...

x/160 +(1020-x)/105 = 8

21x +32640 -32x = 26880 . . . . . multiply by 3360

5760 = 11x . . . . . . . . . add 11x -26880

523.6 ≈ x . . . . . . . . . . divide by 11

Aidan traveled about 523.6 km by train.

3 0

2 years ago

In how many ways can Joseph, Michael, Chris, Katherine, Rebecca and Kyle stand in a line?

Andrews [41]

Answer:

720

Step-by-step explanation:

I got it from Quora

We can use 6 people without repetition, which means we simply have to use counting principle 1.

A number of n objects can be arranged in n! ways.

Total number of combinations possibles = 6! =720

6 0

3 years ago

Alewuya invests a total of $29,000 in two accounts paying 15% and 6% annual interest, respectively. How

Jlenok [28]

2,523 is your investment for 15%
1,611 will be your investment for the 6%

5 0

2 years ago