1^3+3^3+5^3+....to n terms.

To find the sum of the series 1^3 + 3^3 + 5^3 + ... up to n terms, we need to use the formula for the sum of an arithmetic series.

In this series, we can see that the terms are in the form (2k-1)^3, where k represents the position of each term.

Let's break down the steps to find the sum of the series:

1. First, let's find the general term of the series. The general term is given by Tn = (2n-1)^3, where n represents the position of each term in the series.

2. Now, we need to find the sum S of the series up to n terms. We can use the formula for the sum of an arithmetic series, which is given by Sn = (n/2)(a + L), where Sn represents the sum of the series up to n terms, n represents the number of terms, a represents the first term, and L represents the last term.

3. To find the first term a, we substitute n = 1 into the general term equation: a = (2(1) - 1)^3 = (1)^3 = 1.

4. To find the last term L, we substitute n into the general term equation: L = (2n - 1)^3.

5. Now, we have all the information we need to find the sum of the series. Substituting the values into the sum formula, we get:
Sn = (n/2)(a + L) = (n/2)(1 + (2n - 1)^3).

By following these steps and plugging in the appropriate values, you can find the sum of the series 1^3 + 3^3 + 5^3 + ... up to n terms.