Mathematics, 18.08.2019 21:30 hallmansean04

what is the sum of the infinite geometric series? .1/2+1/4+1/8+1/16+

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Answer from: kat1357

$1/2+1/4+1/8+1/16+...=\sum\limits_{n=1}^\infty (\frac{1}{2})^n = \frac{\frac{1}{2}}{1-\frac{1}{2}}=1$

We can see that consecutive fractions are made from 1/2 to consecutive powers. Because we begin with 1/2, n=1
We will infinitely add fractions , hence Lemniscate sign.

In other words you cannot find a number which needs to be added to the geometric series to get "1", therefore the answer is 1. I remember the teacher explaining it this way :)

Answer from: qmsadler01

The sum of the given geometric series is, 1

Step-by-step explanation:

Geometric sequence states that a sequence of numbers that follows a pattern were the next term is found by multiplying by a constant called the common ratio (r).

The sum of the infinite terms of a geometric series is given by:

$S_\infty = \frac{a}{1-r}$ ......[1] ;where

Given the series: $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+.....$

Since, this series is geometric series with constant term(r) = $\frac{1}{2}$

Since,

$\frac{\frac{1}{4}}{\frac{1}{2} } =\frac{2}{4} = \frac{1}{2}$ ,

$\frac{\frac{1}{8}}{\frac{1}{4}} =\frac{4}{8} = \frac{1}{2}$ and so on....

Here, first term(a) = $\frac{1}{2}$

Substitute the values of a and r in [1] we get;

$S_\infty = \frac{\frac{1}{2}}{1-\frac{1}{2}}$ where r = $\frac{1}{2}< 1$

$S_\infty = \frac{\frac{1}{2}}{\frac{2-1}{2}}$

$S_\infty = \frac{\frac{1}{2}}{\frac{1}{2}}$

Simplify:

$S_\infty = 1$

Therefore, the sum of the infinite geometric series is, 1

Answer from: Quest

answer: c ?

step-by-step explanation:

Answer from: Quest

d. $180

step-by-step explanation:

if the dimensions of the room is 9ft by 12 ft, we have the area of the floor as,

$a=9\times 12=108\text {ft^2}$

so the area of the floor is 108 square feet, converting that into square yards we get,

$108 \times 0.1111= 12 \text{yd^2}$

the cost of carpet is $15 per square yard, therefore the cost of carpet for 12 square yards is,

$\$15 \times 12=\$180$

hence, the cost of carpet is $180.

assuming the dimensions of the room be,

15 ft by 12 ft

so, the area of the floor of that room is,

$a = 12 \times 15=180\text {ft}^2$

we can convert 180 square feet into square yards, we know that,

1 sq ft = 0.111 sq yard

so, 180 square feet is,

$180 \times 0.111=20 \text {yd}^2$

so the area of floor is 20 square yards.

now, the cost of carpet is 15 dollars per square yard, therefore the cost of carpet for 20 square yards is,

$\$15 \times 20= \$300$

so, it will cost $300.

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