Describe an infinite geometric series with a beginning value of 2 that converges to 10. What are the first four terms of the sequence?
a{1} = first term of series
∞
Infinite Sum = ∑ a{1} • r^(n – 1) = a{1} ⁄ (1 – r) ... for any geometric series
n =1
Infinite Sum for this problem = 10 = a{1} ⁄ (1 – r) ... a{1} = 2 (given)
10 = 2 ⁄ (1 – r)
r = 0.8 ... common ratio
∞
Infinite Sum = ∑ 2 • (0.8)^(n – 1) ◀◀ (answer)
n =1
a{n} = 2 • (0.8)^(n – 1)
a{1} = 2 • (0.8)^(1 – 1) = 2
a{2} = 2 • (0.8)^(2 – 1) = 1.6
a{3} = 2 • (0.8)^(3 – 1) = 1.28
a{4} = 2 • (0.8)^(4 – 1) = 1.024
a{5} = 2 • (0.8)^(5 – 1) = 0.8192
Infinite Sum = 2 + 1.6 + 1.28 + 1.024 + 0.8192 + . . . + 2 • (0.8)^(n – 1) ****(answer)
correct all the way. Good work.
is this right???
Well, it looks like you've already done the math. But you know what they say, math without a little humor is like a clown without a red nose - it just doesn't quite work! Anyway, here are the first four terms of the infinite geometric series with a beginning value of 2 that converges to 10:
a1 = 2
a2 = 1.6
a3 = 1.28
a4 = 1.024
And don't worry, I won't keep going with the terms, because infinity is just too long of a guest list for my clown car!
An infinite geometric series is a series of numbers in which each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio.
In this case, the beginning value of the series is 2 and the series converges to 10. To find the common ratio, we can use the formula for the sum of an infinite geometric series:
Infinite Sum = a{1} / (1 - r)
Plugging in the given values, we have:
10 = 2 / (1 - r)
Solving for r, we find that the common ratio is 0.8.
Now, we can find the first four terms of the sequence by plugging in the values of n into the formula for the nth term:
a{n} = 2 * (0.8)^(n - 1)
Calculating the first four terms:
a{1} = 2 * (0.8)^(1 - 1) = 2
a{2} = 2 * (0.8)^(2 - 1) = 1.6
a{3} = 2 * (0.8)^(3 - 1) = 1.28
a{4} = 2 * (0.8)^(4 - 1) = 1.024
So, the first four terms of the sequence are 2, 1.6, 1.28, and 1.024.
The infinite sum of the series is the sum of all the terms:
Infinite Sum = 2 + 1.6 + 1.28 + 1.024 + ...
To find the first four terms of the infinite geometric series with a beginning value of 2 and converges to 10, we can use the formula for the nth term of a geometric sequence.
The formula for the nth term of a geometric sequence is given by a{n} = a{1} • r^(n-1), where a{n} is the nth term, a{1} is the first term, r is the common ratio, and n is the term number.
In this case, we are given that the beginning value (a{1}) is 2 and the series converges to 10. To find the common ratio (r), we can use the formula for the sum of an infinite geometric series, which is given by 10 = a{1} / (1 - r).
Using this formula, we can solve for r:
10 = 2 / (1 - r)
10(1 - r) = 2
10 - 10r = 2
10r = 8
r = 0.8
Now that we have the common ratio, we can find the first four terms of the sequence by plugging in the values of n into the formula for the nth term:
a{1} = 2 • (0.8)^(1-1) = 2
a{2} = 2 • (0.8)^(2-1) = 1.6
a{3} = 2 • (0.8)^(3-1) = 1.28
a{4} = 2 • (0.8)^(4-1) = 1.024
So, the first four terms of the sequence are 2, 1.6, 1.28, and 1.024.