Solve for the geometric series of the first 6 terms if the given are : a1=32 r=1/2
32
32 * 1/2
32 * (1/2)^2
and so on
Sure, let's find the sum of the first 6 terms of the geometric series with a1 = 32 and r = 1/2.
The formula for the sum of a geometric series is:
S = a1*(1 - r^n) / (1 - r)
where S is the sum, a1 is the first term, r is the common ratio, and n is the number of terms.
Plugging in the values:
S = 32*(1 - (1/2)^6) / (1 - 1/2)
Simplifying this expression:
S = 32*(1 - 1/64) / (1/2)
S = 32*(63/64) / (1/2)
S = 32*(63/64) * (2/1)
S = 32 * 63
S = 2016
Therefore, the sum of the first 6 terms is 2016.
And if you ask me why I solved it seriously instead of making a joke, well, I'm a Clown Bot, not a mathematician-bot!
To find the sum of the first 6 terms of a geometric series, we will use the formula:
Sn = a1(1 - r^n) / (1 - r)
Where:
Sn = sum of the first n terms
a1 = first term of the series
r = common ratio
n = number of terms
Given:
a1 = 32
r = 1/2
n = 6
Plugging in the values into the formula, we get:
S6 = 32(1 - (1/2)^6) / (1 - 1/2)
Calculating the numerator:
1 - (1/2)^6 = 1 - 1/64 = 63/64
Simplifying the expression:
S6 = 32(63/64) / (1/2)
S6 = (32*63*2) / (64)
S6 = 2016/64
S6 = 31.5
Therefore, the sum of the first 6 terms of the geometric series is 31.5.
To solve for the geometric series of the first 6 terms, we need to use the formula for the sum of a geometric series:
Sn = a * (1 - r^n) / (1 - r)
Where:
Sn represents the sum of the first n terms of the geometric series,
a denotes the first term of the series,
r represents the common ratio between terms, and
n represents the number of terms in the series.
Given that the first term (a1) is 32 and the common ratio (r) is 1/2, we are asked to find the sum of the first 6 terms. Therefore, we have:
a = 32 (first term)
r = 1/2 (common ratio)
n = 6 (number of terms)
Substituting these values into the formula, we have:
S6 = 32 * (1 - (1/2)^6) / (1 - 1/2)
Now, let's calculate the sum:
S6 = 32 * (1 - 1/64) / (1/2)
S6 = 32 * (63/64) / (1/2)
S6 = 32 * (63/64) * (2/1)
S6 = 32 * (63/32)
S6 = 63
Therefore, the sum of the first 6 terms of the geometric series is 63.