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3 years ago

How many quarters can fill a 1 liter bottle?

Mathematics

1 answer:

scoundrel [369]3 years ago

3 0

3090 because I looked up

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Find the general term of {a_n}

Assoli18 [71]

From the given recurrence, it follows that

$a_{n+1} = 2a_n + 1 \\\\ a_{n+1} = 2(2a_{n-1} + 1) = 2^2a_{n-1} + 1 + 2 \\\\ a_{n+1} = 2^2(2a_{n-2}+1) + 1 + 2 = 2^3a_{n-2} + 1 + 2 + 2^2 \\\\ a_{n+1} = 2^3(2a_{n-3} + 1) + 1 + 2 + 2^2 = 2^4a_{n-3} + 1 + 2 + 2^2 + 2^3$

and so on down to the first term,

$a_{n+1} = 2^na_1 + \displaystyle \sum_{k=0}^{n-1}2^k$

(Notice how the exponent on the 2 and the subscript of a in the first term add up to n + 1.)

Denote the remaining sum by S ; then

$S = 1 + 2 + 2^2 + \cdots + 2^{n-1}$

Multiply both sides by 2 :

$2S = 2 + 2^2 + 2^3 + \cdots + 2^n$

Subtract 2S from S to get

$S - 2S = 1 - 2^n \implies S = 2^n - 1$

So, we end up with

$a_{n+1} = 4\cdot2^n + S \\\\ a_{n+1} = 2^2\cdot2^n + 2^n-1 \\\\ a_{n+1} = 2^{n+2} + 2^n - 1 \\\\\implies \boxed{a_n = 2^{n+1} + 2^{n-1} - 1}$

5 0

3 years ago

it cost $20 per hour to bowl and $3 for shoe rental. write a verbal model and algebraic expressions to represent the cost for n

tatyana61 [14]

20n+3
N is hours
20 per hour
+3 for shoe rental

5 0

3 years ago

Which is greater or equal 7/10 or 9/100

Pavlova-9 [17]

9/100 is less because if we multiply 7/10 by 10 (to get an equivalent fraction with a denominator the same as the other fraction [100]) we get 70/100, which is way more than 9/100.

So, 7/10 is greater than 9/100

Hope this helps!

3 0

3 years ago

If rectangle is 8 feet long and (7+x) feet wide. what is the area of the rectangle in square feet?

Ksivusya [100]

if the length is 8ft and the width is (7+x) then the area would be 56+8x, if x is unknown.

5 0

4 years ago

What is the area of a circle. Using 3.14 for. A circle with diameter of 8.4 ft.

ira [324]

Hey there! I'm happy to help!

First, let's find the radius. The radius is half of the diameter.

8.4/2=4.2

Here's how to find the area of a circle.

Square the radius.

4.2²=17.64

And multiply by pi (3.14 in our case)

17.64(3.14)=55.3896

We round to the nearest tenth, giving us an area of 55.4 ft²

Have a wonderful day! :D

5 0

3 years ago