The given series is an arithmetic sequence with a common difference of 4. We can write the sum of the first n terms of an arithmetic sequence as:
S_n = (n/2)(a_1 + a_n)
where S_n is the sum of the first n terms, a_1 is the first term, and a_n is the nth term.
In this case, we are given that:
1 + 5 + 9 + 13 +...+ x = 780
To use the formula for the sum of an arithmetic sequence, we need to find the first term and the nth term. The first term is 1, and we can find the nth term using the formula for the nth term of an arithmetic sequence:
a_n = a_1 + (n-1)d
where d is the common difference. In this case, d = 4, so:
a_n = 1 + (n-1)(4) = 4n - 3
Now we can substitute a_1 and a_n into the formula for the sum of the first n terms:
S_n = (n/2)(a_1 + a_n) = (n/2)(1 + 4n - 3) = (n/2)(4n - 2)
We want to find the value of x such that the sum of the series is 780, so we can set S_n equal to 780 and solve for n:
(n/2)(4n - 2) = 780
2n(2n - 1) = 780
4n^2 - 2n - 780 = 0
We can use the quadratic formula to solve for n:
n = (-(-2) ± sqrt((-2)^2 - 4(4)(-780))) / (2(4))
n = (2 ± sqrt(2^2 + 4(4)(780))) / 8
n = (2 ± sqrt(6242)) / 8
We can ignore the negative solution because n must be a positive integer. Evaluating the positive solution gives:
n = (2 + sqrt(6242)) / 8
n ≈ 10.75
Since n must be a positive integer, we round up to the nearest integer to get n = 11. Therefore, the 11th term of the series is:
a_11 = 4n - 3 = 4(11) - 3 = 43
Therefore, the value of x is 43.