Question

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.

Answer 1

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.