Describe an infinite geometric series with a beginning value of 2 that converges to 10. What are the first 4 terms of the series?

Last question and my mind is fried, can someone please help me? Thanks

S ∞ =


∑ = a * r ^ n =  a  ⁄ ( 1   –  r )
n=1

In this case a = 2 , infinite sum = 10  so:

S ∞ = 10 = a  ⁄ ( 1   –  r )

10 = 2  ⁄ ( 1   –  r ) Multiply both sides by 1 - r

10 * ( 1   –  r ) = 2 

10 * 1   –  10 * r = 2 

10 - 10 r = 2 Subtract 10 to both sides

10 - 10 r - 10 = 2 - 10

- 10 r = - 8 Divide both sides by - 10

- 10 r / - 10 = - 8 / - 10

r = 0.8

The n-th term of a geometric series with initial value a and common ratio r is given by:

an = a * r ^ ( n − 1 )

a1 = a * r ^ ( n − 1 ) = 2 * 0.8 ^ ( 1 − 1 ) = 2 * 0.8 ^ 0 = 2 * 1 = 2

a2 = a * r ^ ( n − 1 ) = 2 * 0.8 ^ ( 2 − 1 ) = 2 * 0.8 = 1.6

a3 = a * r ^ ( n − 1 ) = 2 * 0.8 ^ ( 3 − 1 ) = 2 * 0.8 ^ 2 = 2 * 0.64 = 1.28

a4 = a * r ^ ( n − 1 ) = 2 * 0.8 ^ ( 4 − 1 ) = 2 * 0.8 ^ 3 = 2 * 0.512 = 1.024

Sure, I can help you with that!

An infinite geometric series is a sequence of numbers where each term is found by multiplying the previous term by a constant ratio.

In this case, the beginning value (first term) is 2 and the series converges to 10.

To find the common ratio, we can divide the second term by the first term, and similarly, the third term by the second term, and so on.

Let's assume the common ratio is denoted as "r".

Since the series converges to 10, we can write the equation:

2 + 2r + 2r^2 + 2r^3 + ...

To find the common ratio "r", we can divide the second term by the first term:

(2r) / 2 = r

Similarly, dividing the third term by the second term:

(2r^2) / (2r) = r

Setting these two equations equal to each other:

r = r^2

Dividing both sides by "r":

1 = r

So, we have found that the common ratio "r" is 1.

Now that we know the common ratio, we can find the first 4 terms of the series:

1st term: 2
2nd term: 2 * 1 = 2
3rd term: 2 * 1 * 1 = 2
4th term: 2 * 1 * 1 * 1 = 2

Therefore, the first 4 terms of the infinite geometric series are: 2, 2, 2, 2.

Of course, I can help you with that. An infinite geometric series is a sequence of numbers in which each term is found by multiplying the preceding term by a constant ratio. In this case, we have a series with a beginning value of 2 that converges to 10.

To find the constant ratio, we divide any term (let's call it "a") by its preceding term (let's call it "r"). So we have a/r = 10/2, or a/r = 5. This means that the constant ratio (r) is 5.

Now that we know the constant ratio, we can find the first few terms of the series. The general formula for an infinite geometric series is a, ar, ar^2, ar^3, ..., where a is the first term and r is the common ratio.

In this case, the first term (a) is 2 and the common ratio (r) is 5. So the first four terms of the series would be:
2, 2 x 5, 2 x 5^2, 2 x 5^3

Simplifying these terms, we get:
2, 10, 50, 250

Therefore, the first four terms of the infinite geometric series with a beginning value of 2 and converging to 10 are 2, 10, 50, and 250.