The sum of the series $1^{3}+3^{3}+5^{3}+....$ upto $20$ terms is
- $319600$
- $321760$
- $306000$
- $347500$
The series is 1^3, 3^3, 5^3, ... The n-th term is (2n-1)^3. The sum is sum_{n=1}^{20} (8n^3 - 12n^2 + 6n - 1). Using summation formulas: 8 * (20*21/2)^2 - 12 * (20*21*41/6) + 6 * (20*21/2) - 20 = 8 * 44100 - 12 * 2870 + 1260 - 20 = 352800 - 34440 + 1240 = 319600.
The series consists of cubes of odd numbers, so its general term is (2n - 1)^3, which expands to 8n^3 - 12n^2 + 6n - 1. Using the standard formulas for the sum of n terms for these powers, the sum is 8[n(n+1)/2]^2 - 12[n(n+1)(2n+1)/6] + 6[n(n+1)/2] - n. For n = 20, these individual sums evaluate to 8(44100), 12(2870), 6(210), and 20, resulting in 352800 - 34440 + 1260 - 20. The final result of this calculation is 319600.