Find the number of terms in the geometric series, 5+15+…+3645
[(3645 - 5) / (15 - 5)] + 1
In a GP,
term(n) = a r^(n-1)
in yours, a = 5, r = 3
3645 = 5(3^(n-1) )
729 = 3^(n-1)
3^6 = 3^(n-1)
n-1 = 6
n = 7
There are 7 terms
in case you want the sum
sum(7) = a(r^7 - 1)/(r-1)
= 5(3^7 - 1)/2 = 5465
To find the number of terms in a geometric series, we need to identify the first term, the common ratio, and the last term.
In this case, the first term is 5, and we need to find the last term.
Let's find the common ratio first, which we can do by dividing any term by its previous term.
15 ÷ 5 = 3
So, the common ratio is 3.
Now, let's find the last term.
We can do this by using the formula for the nth term of a geometric series:
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 number of terms.
Plugging in the value,
3645 = 5 * 3^(n-1)
Simplifying the equation:
729 = 3^(n-1)
Now, let's express both sides of the equation with the same base so we can solve for n.
729 = 3^6
So, n-1 = 6.
Adding 1 to both sides, we get:
n = 7.
Therefore, the geometric series 5 + 15 + ... + 3645 has 7 terms.
To find the number of terms in a geometric series, we need to determine the common ratio first. In a geometric series, each term is obtained by multiplying the previous term by a common ratio.
In this case, let's examine the first few terms of the given series to find the common ratio:
5, 15, ...
To get from 5 to 15, we multiply by 3. Therefore, the common ratio (r) is 3.
The formula to find the number of terms in a geometric series is given by:
n = logᵣ((termₙ₊₁)/(first term)) + 1
Here, termₙ₊₁ represents the last term in the series (3645), and the first term is 5.
Substituting these values into the formula, we get:
n = log₃((3645/5)) + 1
To evaluate this expression, we can use logarithmic rules or a calculator:
n ≈ log₃(729) + 1
n ≈ 6 + 1
n ≈ 7
Therefore, the given geometric series has 7 terms.