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

Answers

Answer 1

Answer:

Answer:

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:

$S_\infty = (a)/(1-r)$ ......[1] ;where $0<r<1$

Given the series: $(1)/(2)+(1)/(4)+(1)/(8)+(1)/(16)+.....$

Since, this series is geometric series with constant term(r) = $(1)/(2)$

Since,

$((1)/(4))/((1)/(2) ) =(2)/(4) = (1)/(2)$ ,

$((1)/(8))/((1)/(4)) =(4)/(8) = (1)/(2)$ and so on....

Here, first term(a) = $(1)/(2)$

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

$S_\infty = ((1)/(2))/(1-(1)/(2))$ where r = $(1)/(2)< 1$

$S_\infty = ((1)/(2))/((2-1)/(2))$

$S_\infty = ((1)/(2))/((1)/(2))$

Simplify:

$S_\infty = 1$

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

Answer 2

Answer: $1/2+1/4+1/8+1/16+...=\sum\limits_(n=1)^\infty ((1)/(2))^n = ((1)/(2))/(1-(1)/(2))=1$

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 :)

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Peter & Alex share a lottery win of £4450 in the ratio 4 : 1. Peter then shares his part between himself, his wife & their son in the ratio 1 : 6 : 3. How much more does his wife get over their son?

Answers

Answer:

His wife gets $1068 more than his son.

Step-by-step explanation:

Peter and Alex:

Peter gets $(4)/(4+5) = (4)/(5)$ of the total amount.

$(4*4450)/(5) = 3560$

Peter gets $3560. He divides with his wife and son.

Peter gets $(1)/(1+6+3) = (1)/(10)$ of the amount, the wife gets $(6)/(1+6+3) = (6)/(10)$ and the son gets $(3)/(1+6+3) = (3)/(10)$

Wife:

$(6*3560)/(10) = 2136$

His wife gets $2136

Son:

$(3*3560)/(10) = 1068$

How much more does his wife get over their son?

2136 - 1068 = 1068

His wife gets $1068 more than his son.

In a 30-60-90 right triangle, what is the value for x? Check my answer?

Answers

Your answer is not visible.

The length of 'x' is 5 times the square root of 3.

4^(x-1) + 4^x + 4^(x+1)= 21

Answers

Answer: x = 1

Step-by-step explanation:

isolate the variable by dividing both sides by factors that don't contain the variable.

Which of the following shows 27/54 written in prime factored form to help in reducing the fraction to simplest form? A. 9x3/9x6
B. 27x1/27x2
C. 3x3x3/3x3x3x2
D. 1/2