Find the number of terms in the geometric series 5+.15+...+3645
I suspect that .15 is supposed to be 15, since 15/5 = 3 and 3645/5 = 729 = 3^6
so there are 7 terms
This makes no sense.
Looking at your first two terms, the common ratio is .15/5 = .03
so the terms would be getting smaller quite rapidly.
No way a term can be 3645
Is there a missing decimal somewhere??
To find the number of terms in a geometric series, we first need to determine the common ratio. In a geometric series, each term is obtained by multiplying the previous term by the common ratio.
Let's analyze the given series: 5, 0.15, ...
To find the common ratio, we divide each term by its previous term:
0.15/5 = 0.03
Now we have the common ratio, which is 0.03. We need to find the number of terms in the series, given the first term (5) and the last term (3645).
To find the number of terms using the first term, last term, and the common ratio, we can use the formula:
n = log(last term / first term) / log(common ratio),
where n is the number of terms.
Applying this formula to our problem, we have:
n = log(3645 / 5) / log(0.03)
Using a calculator, we can compute this value:
n ≈ log(729) / log(0.03) ≈ 3 / (-1.5229) ≈ -1.9727
However, a negative number of terms is not possible, so we need to take the absolute value of the result:
n ≈ |-1.9727| ≈ 1.9727
Since we cannot have a fraction of a term, we round up to the nearest whole number:
n ≈ 2
Therefore, the geometric series 5, 0.15, ... , 3645 has approximately 2 terms.