Identify n(S) for the given set S = { 5, 10, 15, 20, 25, 30, . . .}
To identify n(S) for the given set S = {5, 10, 15, 20, 25, 30, . . .}, we need to determine the number of elements in the set.
To find the number of elements in S, we need to first understand the pattern in the set. Looking at the given set, we can observe that it consists of multiples of 5, starting from 5 and increasing by 5 each time. This means that the set S contains an infinite number of elements.
In mathematical terms, we can express this as follows:
S = {5, 10, 15, 20, 25, 30, ...}
The pattern suggests that we can generate any element in the set using the formula:
an = a1 + (n-1)d
where:
an is the nth term
a1 is the first term
d is the common difference
In this case, the first term (a1) is 5, and the common difference (d) is 5.
Using the formula, we can find the nth term, which represents the number of elements (n) in the set:
an = a1 + (n-1)d
We want to find the nth term where an is greater than any number we choose, so we can let an be any large number, say M.
M = a1 + (n-1)d
Simplifying this equation, we get:
M = 5 + 5(n-1)
Now, let's solve for n:
M = 5 + 5n - 5
5n = M
n = M/5
From this equation, we can see that the number of elements in the set (n) is equal to the chosen large number (M) divided by 5.
Therefore, n(S) = M/5.
It's important to note that since the set S contains an infinite number of elements, n(S) will also be infinity.