Find the sum of the following series 8+12+18+as far the 6th term
729/8
8+12+18+26+36+48
60.75
To find the sum of a series, you need to first determine the pattern or relationship between the terms. In this case, we are given the series 8, 12, 18, and so on.
By observing the terms, we can see that each term is obtained by multiplying the previous term by a common ratio. In this case, the common ratio is 1.5, since 8 * 1.5 = 12, 12 * 1.5 = 18, and so on.
To find the sixth term, we can use the formula for the nth term of a geometric sequence:
tn = a * r^(n-1)
Where:
tn is the nth term,
a is the first term, and
r is the common ratio.
In this case, a = 8, r = 1.5, and n = 6.
t6 = 8 * 1.5^(6-1)
= 8 * 1.5^5
= 8 * 1.5^5
= 8 * 7.59375
= 60.75
So, the sixth term of this series is 60.75.
Now that we have found the sixth term, we can find the sum of the first six terms using the formula for the sum of a geometric series:
Sn = a * (1 - r^n) / (1 - r)
Where:
Sn is the sum of the first n terms.
In this case, a = 8, r = 1.5, and n = 6.
S6 = 8 * (1 - 1.5^6) / (1 - 1.5)
= 8 * (1 - 1.5^6) / (-0.5)
= 8 * (1 - 1.5^6) / -0.5
= 8 * (1 - 7.59375) / -0.5
= 8 * (-6.59375) / -0.5
= -52.75
Therefore, the sum of the series up to the sixth term is -52.75.