Find the sum 5/3+9/3+17/3+33/3+...
I answered infinity, but that just seems too simple. Thanks for your help!
If that is an infinite series, the sum is infinity.
5x+2>1
To find the sum of the infinite series 5/3 + 9/3 + 17/3 + 33/3 + ..., we can observe that each term in the series is obtained by multiplying the previous term by 2 and adding 1. Let's denote the terms of the series as a₁, a₂, a₃, a₄, and so on.
The first term, a₁, is given as 5/3.
The second term, a₂, can be obtained by multiplying the first term by 2 and adding 1:
a₂ = (5/3) * 2 + 1 = 11/3.
Similarly, we can calculate the third term, a₃, by multiplying the second term by 2 and adding 1:
a₃ = (11/3) * 2 + 1 = 23/3.
Continuing this pattern, we find the fourth term, a₄:
a₄ = (23/3) * 2 + 1 = 47/3.
Therefore, the terms of the series can be written as:
a₁ = 5/3,
a₂ = 11/3,
a₃ = 23/3,
a₄ = 47/3,
and so on.
Now, to find the sum of the series, we can use the formula for the sum of an infinite geometric series when the absolute value of the common ratio r is less than 1. In this case, the common ratio is 2.
The formula for the sum of an infinite geometric series is given by:
S = a / (1 - r),
where S represents the sum of the series, a represents the first term, and r represents the common ratio.
Applying this formula to our series, we have:
S = (5/3) / (1 - 2).
Since the absolute value of the common ratio r is greater than 1 (r = 2), the sum of this infinite series does not exist. In other words, the sum diverges to infinity.
Therefore, the answer is indeed infinity in this case.