The correct answer is option (B):
810
Let's break down how to solve this problem involving an arithmetic progression (AP).
First, let's establish some variables. Let 'a' represent the first term of the AP, and let 'd' represent the common difference between consecutive terms.
We're given two pieces of information:
1. The second term is 15. The formula for the nth term of an AP is a + (n-1)d. Therefore, the second term (n=2) is a + (2-1)d = a + d. So we have the equation: a + d = 15.
2. The fifth term is double the first term. The fifth term (n=5) is a + (5-1)d = a + 4d. Double the first term is 2a. Therefore, we have the equation: a + 4d = 2a.
Now we have a system of two equations with two variables:
* Equation 1: a + d = 15
* Equation 2: a + 4d = 2a
Let's solve this system. We can rearrange Equation 2 to make it easier to work with. Subtract 'a' from both sides: 4d = a. Now we know that a = 4d.
Substitute this value of 'a' (4d) into Equation 1: 4d + d = 15. This simplifies to 5d = 15. Divide both sides by 5: d = 3.
Now that we know d = 3, we can substitute it back into a = 4d to find 'a': a = 4 * 3 = 12.
So, the first term (a) is 12, and the common difference (d) is 3.
Finally, we need to find the sum of the first 20 terms of the AP. The formula for the sum (S_n) of the first n terms of an AP is: S_n = (n/2) * [2a + (n-1)d]
In our case, n = 20, a = 12, and d = 3. Substituting these values:
S_20 = (20/2) * [2(12) + (20-1)3]
S_20 = 10 * [24 + 19 * 3]
S_20 = 10 * [24 + 57]
S_20 = 10 * 81
S_20 = 810
Therefore, the sum of the first 20 terms of the series is 810.