Find the sum of the geometric series. 1+3+9+…+2187
To find the sum of a geometric series, we can use the formula:
S = a * (1 - r^n) / (1 - r)
where:
S = sum of the geometric series
a = first term
r = common ratio
n = number of terms
In this case, the first term (a) is 1, the common ratio (r) is 3, and the last term is 2187. To find the number of terms (n), we can use the formula for the nth term of a geometric series:
an = a * r^(n-1)
Plugging in the values, we get:
2187 = 1 * 3^(n-1)
2187 = 3^(n-1)
We can rewrite 2187 as 3^7, so:
3^7 = 3^(n-1)
7 = n - 1
n = 8
Now, we can plug in the values of a, r, and n into the sum formula:
S = 1 * (1 - 3^8) / (1 - 3)
S = 1 * (1 - 6561) / -2
S = 1 * (-6560) / -2
S = -6560 / -2
S = 3280
Therefore, the sum of the geometric series 1 + 3 + 9 + ... + 2187 is 3280.