If the seventh term of a geometric series is 192, and the first term is 3, find the sum of the first 8 terms of the series
ar^6 = 192 , a=3
r^6 = 192/3 = 64
r = 2
sum(8) =a(r^8 - 1)/(r-1)
= 3(256-1)/(2-1) = 765
Mathe matics
To find the sum of the first 8 terms of a geometric series, we need to know the common ratio of the series.
The formula for the nth term of a geometric series is:
an = a * r^(n-1)
where:
an = the nth term of the series,
a = the first term of the series, and
r = the common ratio of the series.
Given that the seventh term (a7) is 192 and the first term (a) is 3, we can substitute these values into the formula to find the common ratio (r):
192 = 3 * r^(7-1)
192 = 3 * r^6
Now let's solve for r:
r^6 = 192/3
r^6 = 64
r = ∛(64)
r = 4
We have found that the common ratio (r) is 4.
Now, we can find the sum of the first 8 terms of the series using the formula:
Sn = a * (1 - r^n) / (1 - r)
where:
Sn = the sum of the first n terms of the series.
Substituting the given values into the formula:
S8 = 3 * (1 - 4^8) / (1 - 4)
S8 = 3 * (1 - 65536) / (-3)
S8 = 3 * (-65535) / (-3)
S8 = 196605
Therefore, the sum of the first 8 terms of the series is 196605.