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2 years ago

8

Four times The sum of 7 and w

1 answer:

Ray Of Light [21]2 years ago

8 0

Answer:

28

Step-by-step explanation:

$7+7+7+7 = 14+7+7 = 21+7 = 28$

O sino:

$7*4 = 28$

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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

Help please..!!!!!!!!!

valina [46]

Answer:

i think b and d

Step-by-step explanation:

3 0

2 years ago

What is the value of f(-1) when f(x)=2x+2? F(-1)=?

igomit [66]

Here is your function:

$f(x)=2x+2$

You're looking for f(-1). To get your answer, you have to replace all the x's with -1.

Here is your new equation:

$f(-1) = 2(-1)+2$

Multiply 2 by -1

$2 \times -1 = -2$

<em>Remember:</em>

MULTIPLYING

Positive(+) times negative(-) is equal to negative

Positive(+) times positive(+) is equal to positive

Negative(-) times negative(-) is equal to positive

Negative(-) times positive(+) is equal to negative

DIVIDING

Positive(+) divided by positive(+) is equal to positive

Positive(+) divided by negative(-) is equal to negative

Negative(-) divided by positive(+) is equal to negative

Negative(-) divided by negative(-0 is equal to positive

Since you're multiplying a negative number by a positive number, the answer will be negative.

Here is your new equation:

$f(-1) = -2+2$

Add -2 and 2:

$-2+2 = 0$

That means your answer is 0

↓

$f(-1)=0$

<em>If you have any questions, feel free to ask in the comments! :)</em>

5 0

2 years ago

Georgia bought a packge of of 14 dollies to use in her valentines.She used 5/7 of the package to make her in valentines. How man

max2010maxim [7]

Answer: Georgia used 10 dollies and she has left 4 dollies .

Step-by-step explanation:

Given: Georgia bought a package of of 14 dollies to use in her valentines.

Therefore, the number of dollies in a package = 14

If she used $\frac{5}{7}$ of the package to make her in valentines.

The number of dollies she used = $\frac{5}{7}\times14=5\times2=10$

The number of dollies left = $14-10=4$

Hence, Georgia used 10 dollies and she has left 4 dollies .

5 0

2 years ago

g.o is going on a hike in the mountains on a hot summer day. he has 4/5 gallon of water that splits into 4 bottles. what fractio

Pani-rosa [81]

If he splits it into 4 bottles you have to convert 4/5 into a decimal and you would get .8 then
.8 divided by 4 would be .2 two tenths so there would be. two tenths of a gallon in each bottle
.2

4 0

3 years ago

Read 2 more answers