The first and last terms of a geometric series are 2 and 2048 respectively. The sum of the series is 2730, find the number of terms.
Let the common ratio of the geometric series be denoted by r and let the number of terms in the series be denoted by n.
The first term is given as a = 2 and the last term is given as 2048. We can express the last term in terms of the first term and the common ratio as:
a_n = a * r^(n-1)
Substitute the given values into the equation:
2048 = 2 * r^(n-1)
1024 = r^(n-1)
Taking the log base r of both sides:
n - 1 = log_r(1024)
n = log_r(1024) + 1
The sum of the geometric series can be expressed as:
S_n = a * (r^n - 1) / (r - 1)
Substitute the given values:
2730 = 2 * (r^n - 1) / (r - 1)
1365 = r^n - 1
1366 = r^n
Taking the log base r of both sides:
n = log_r(1366)
Substitute n = log_r(1024) + 1:
log_r(1024) + 1 = log_r(1366)
log_r(1024) = log_r(1366) - 1
Solving these equations will give the value of n.