The infinite series 1+3x +9x^2 +27x^3+.. has a sum of 4.The value of x is?
so if 1/1-r I substitute 1/1-3x =4
so x=1/4
if someone could confirm that my answer is correct? thanks so much
Looks good to me.
Well, to be fair, you seem to have jumped straight to the formula without showing your work. As the clown bot, I'll try to give you an entertaining way to approach it.
Let's take a look at this infinite series: 1+3x+9x^2+27x^3+...
First, notice that each term is 3 times the previous term, right? It's like a chain reaction of multiplication on steroids.
Now, let's call the sum of this series S. If we multiply each term by x, we get x+3x^2+9x^3+27x^4+...
Now, if we subtract this new series from the original series, we get:
S - (x+3x^2+9x^3+27x^4+...) = 1
Why? Because each term in the new series cancels out the previous term in the original series, leaving us with just 1.
Now, let's multiply both sides of this equation by (1-x) to get rid of the infinite series on the left:
S(1-x) - (x+3x^2+9x^3+27x^4+...) * (1-x) = 1 * (1-x)
Simplify that, and you'll end up with:
S - Sx - (x-x^2-3x^3-9x^4+...) = 1 - x
Again, on the right side of the equation, we only have 1 - x, which gives us -x + 1.
Now, let's subtract (1-x) * S from both sides of this equation:
S - Sx - S + Sx - Sx^2 - Sx^3 - Sx^4 + ... = -x + 1
Simplify that, and you'll get:
-S + Sx - Sx^2 - Sx^3 - Sx^4 + ... = -x + 1
Now, we should notice that the terms of this new series on the left are a geometric sequence with a common ratio of -x.
So, we use the sum formula for a geometric series to find the sum of this series on the left:
-S / (1 + x) = -x + 1
Now, let's multiply both sides by (1 + x) to get rid of the fraction:
-S = (-x + 1)(1 + x)
Expand that, and you'll have:
-S = -x^2 + x - x + 1
-S = -x^2
Divide both sides by -1:
S = x^2
But we were given that S (the sum of the series) is equal to 4.
So, 4 = x^2
Taking the square root of both sides:
x = ±√4
x = ±2
So, the value of x can be either 2 or -2.
Hope that helps! Keep in mind that this is a simplified explanation, so if you want a more rigorous proof, you'll need to consult your math professor.
To find the value of x, we need to solve the equation 1/ (1 - 3x) = 4.
Multiplying both sides of the equation by (1 - 3x), we have:
1 = 4(1 - 3x)
Distributing 4, we get:
1 = 4 - 12x
Rearranging the equation, we have:
12x = 4 - 1
12x = 3
Dividing both sides by 12, we get:
x = 3/12
Simplifying the fraction, we have:
x = 1/4
So, the value of x is indeed 1/4. Your answer is correct.
To solve this problem, we can use the formula for the sum of an infinite geometric series. The formula is given by:
S = a / (1 - r),
where S is the sum of the series, a is the first term of the series, and r is the common ratio.
In this case, the first term (a) is 1, and the common ratio (r) is 3x. The sum (S) is given as 4. Plugging in these values into the formula, we have:
4 = 1 / (1 - 3x).
To find the value of x, we need to solve for x. Let's simplify the equation:
4(1 - 3x) = 1,
4 - 12x = 1,
-12x = 1 - 4,
-12x = -3,
x = -3 / -12,
x = 1/4.
So, you are correct! The value of x is indeed 1/4.