Question: 2. $\sum_{j=2}^{6}(j^3-1)$
1. **State the problem:** Calculate the summation $$\sum_{j=2}^{6}(j^3-1)$$ which means we sum the values of the expression $$j^3 - 1$$ for $$j$$ from 2 to 6 inclusive.
2. **Formula and rules:** The summation notation $$\sum$$ means to add up all terms for each integer $$j$$ in the range. Here, each term is $$j^3 - 1$$.
3. **Calculate each term:**
- For $$j=2$$: $$2^3 - 1 = 8 - 1 = 7$$
- For $$j=3$$: $$3^3 - 1 = 27 - 1 = 26$$
- For $$j=4$$: $$4^3 - 1 = 64 - 1 = 63$$
- For $$j=5$$: $$5^3 - 1 = 125 - 1 = 124$$
- For $$j=6$$: $$6^3 - 1 = 216 - 1 = 215$$
4. **Sum all terms:**
$$7 + 26 + 63 + 124 + 215 = 435$$
5. **Final answer:**
$$\sum_{j=2}^{6}(j^3-1) = 435$$