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Mekhanik [1.2K]
3 years ago
15

Which of the following geometric series converges?

Mathematics
1 answer:
Artist 52 [7]3 years ago
8 0

All three series converge, so the answer is D.

The common ratios for each sequence are (I) -1/9, (II) -1/10, and (III) -1/3.

Consider a geometric sequence with the first term <em>a</em> and common ratio |<em>r</em>| < 1. Then the <em>n</em>-th partial sum (the sum of the first <em>n</em> terms) of the sequence is

S_n=a+ar+ar^2+\cdots+ar^{n-2}+ar^{n-1}

Multiply both sides by <em>r</em> :

rS_n=ar+ar^2+ar^3+\cdots+ar^{n-1}+ar^n

Subtract the latter sum from the first, which eliminates all but the first and last terms:

S_n-rS_n=a-ar^n

Solve for S_n:

(1-r)S_n=a(1-r^n)\implies S_n=\dfrac a{1-r}-\dfrac{ar^n}{1-r}

Then as gets arbitrarily large, the term r^n will converge to 0, leaving us with

S=\displaystyle\lim_{n\to\infty}S_n=\frac a{1-r}

So the given series converge to

(I) -243/(1 + 1/9) = -2187/10

(II) -1.1/(1 + 1/10) = -1

(III) 27/(1 + 1/3) = 18

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Find the area of the shaded polygons:
Marina86 [1]

Answer:

6.5 square units

Step-by-step explanation:

There are MANY ways to do this problem, most of them involving breaking it into 3 or four shapes.  But this one is easier if you do most of it by taking out the squares and half squares.  

Squares - 3

Half squares - 4 (totals 2 squares)

Triangle - 1 - 1/2bh = 1/2x1x3 = 1.5

TOTAL = 3+2+1.5=6.5

5 0
3 years ago
-3(-7) I need this asap pls help me
7nadin3 [17]

Hi there! Hopefully this helps!

-------------------------------------------------------------------------------------------------

<h2>Answer: \boxed{21}</h2><h2 /><h2>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~</h2>

When you multiply two negatives (-, -,), you are gonna get a positive(+)

<em>[Like making a cross out of two lines].</em>

-3(-7) = +3(+7)

<h2>Since 3(7) normally <u>equals 21</u>, that would be the answer for this case.</h2>

3 0
3 years ago
Read 2 more answers
A building is composed of three rooms as shown. What
Artemon [7]
Ignore the top pat of the picture that’s my government homework. The answer is 104ft.

5 0
3 years ago
Let SSS represent the number of sunflower plants and LLL represent the number of lily plants Ezra's water supply can water.
Fantom [35]

Answer:

Ezra can water at most approximately 8 sunflower plants with the remaining amount of water.

Step-by-step explanation:

Given the inequality function

0.7S+0.5L≤11 where S represent the number of sunflower plants and L represent the number of lily plants Ezra's water supply can water, if Ezra waters 10 lily plants, then we can calculate the maximum amount of sunflower plant that he can water with the remaining amount of water by simply substituting L = 10 into the inequality function as shown;

0.7S+0.5L≤11

0.7S+0.5(10)≤11

0.7S+5≤11

Taking 5 to the other side:

0.7S≤11-5

0.7S≤6

S≤6/0.7

S≤8.57

This shows that Ezra can water at most approximately 8 sunflower plants with the remaining amount of water.

3 0
3 years ago
Bowser ate 4 1/2 pounds of dog food. That amount is 3/4 of an entire bag of dog food. How many pound of dog food were originally
GalinKa [24]
Divide number of pounds eaten by the portion of the bag eaten.

= 4 1/2 pounds ÷ 3/4
= 4.5 ÷ 0.75
= 6 pound bag

CHECK:
multiply total pounds of bag by portion eaten
= 6 pound bag * 3/4
= 4 1/2 pounds eaten

ANSWER: The bag was 6 pounds of dog food.

Hope this helps! :)
8 0
3 years ago
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