Geometric series: Finite and infinite
n=5
a1=243
r=2/3
i=1
This is the equation
243(2/3)^i
My answer is 633
n=infinity sign
a1=5
r=-1/4
i=1
The equation
5(-1/4)^i-1
I think its undefined
a1=243
r = 2/3
an = 243(2/3)^(n-1)
a5 = 48
s = a/(1-r) = 5/(5/4) = 4
To find the sum of a geometric series, you can use the formula:
S = a1 * (1 - r^n) / (1 - r)
For a finite geometric series, where you have a specific number of terms (n), you can substitute the values of a1, r, and n into the equation to find the sum (S). Let's calculate the sum for the given values:
n = 5
a1 = 243
r = 2/3
Substituting the values into the formula:
S = 243 * (1 - (2/3)^5) / (1 - 2/3)
Simplifying the equation:
S = 243 * (1 - 32/243) / (1/3)
S = 243 * (211/243) / (1/3)
S = 243 * (211/243) * (3/1)
Canceling out common factors:
S = 211 * 3
S = 633
Therefore, the sum of the finite geometric series with n = 5, a1 = 243, and r = 2/3 is equal to 633.
For the infinite geometric series, the sum can be calculated if the absolute value of the ratio (r) is less than 1. If the absolute value of r is greater than or equal to 1, the series does not have a sum and is considered divergent.
Let's calculate the sum for the given values:
n = ∞ (infinity sign)
a1 = 5
r = -1/4
If the absolute value of r is less than 1, we can proceed. In this case, the absolute value of r is 1/4, which is less than 1. So, we can calculate the sum.
Substituting the values into the formula:
S = a1 / (1 - r)
S = 5 / (1 - (-1/4))
Simplifying the equation:
S = 5 / (1 + 1/4)
S = 5 / (5/4)
S = 5 * (4/5)
S = 4
Therefore, the sum of the infinite geometric series with n = ∞, a1 = 5, and r = -1/4 is equal to 4. It is not undefined; it has a specific value.