The given series is 2, 6, 18, ..., 4374. We need to find the sum of this series.

To find the sum of this series, we can use the formula for the sum of a geometric series:

Sn = a * (r^n - 1) / (r - 1)

Where:
Sn = sum of the series
a = first term of the series
r = common ratio of the series
n = number of terms in the series

Let's analyze the given series step by step:

Step 1: Finding the common ratio (r)
To find the common ratio (r), we divide each term of the series by the previous term.

6 / 2 = 3
18 / 6 = 3
...
4374 / 1458 = 3

So, the common ratio (r) is 3.

Step 2: Finding the number of terms (n)
To find the number of terms (n), we need to find the exponent (power) to which the common ratio (3) is raised to get the last term (4374).

3^n = 4374
Taking the logarithm on both sides:
n * log(3) = log(4374)
n = log(4374) / log(3)

Using a calculator, we find that n is approximately 7.9998. Since the number of terms must be a positive integer, we round up n to 8.

So, the number of terms (n) is 8.

Step 3: Finding the first term (a)
The first term (a) is given as 2.

Step 4: Calculating the sum (Sn)
Using the formula for the sum of a geometric series:

Sn = a * (r^n - 1) / (r - 1)
Sn = 2 * (3^8 - 1) / (3 - 1)
Sn = 2 * (6561 - 1) / 2
Sn = (2 * 6560) / 2
Sn = 6560

So, the sum of the series 2, 6, 18, ..., 4374 is 6560.

It is a simple gp with common ratio 3 and no of terms 8 apply the formula and get the answer