4) $$ \sum_{i=1}^{10}\left(2 i^{3}+5 i\right) $$

Question

UpStudy AI · Accepted Answer

To evaluate the sum:

$$
\sum_{i=1}^{10}\left(2i^{3} + 5i\right)
$$

we can break it down into two separate sums:

$$
\sum_{i=1}^{10}\left(2i^{3} + 5i\right) = 2\sum_{i=1}^{10}i^{3} + 5\sum_{i=1}^{10}i
$$

**Step 1: Calculate $$\sum_{i=1}^{10}i^{3}$$**

The sum of the cubes of the first $$n$$ natural numbers is given by:

$$
\sum_{i=1}^{n}i^{3} = \left(\frac{n(n + 1)}{2}\right)^{2}
$$

For $$n = 10$$:

$$
\sum_{i=1}^{10}i^{3} = \left(\frac{10 \times 11}{2}\right)^{2} = (55)^{2} = 3025
$$

**Step 2: Calculate $$\sum_{i=1}^{10}i$$**

The sum of the first $$n$$ natural numbers is:

$$
\sum_{i=1}^{n}i = \frac{n(n + 1)}{2}
$$

For $$n = 10$$:

$$
\sum_{i=1}^{10}i = \frac{10 \times 11}{2} = 55
$$

**Step 3: Combine the results**

Substitute the calculated sums back into the original equation:

$$
2\sum_{i=1}^{10}i^{3} + 5\sum_{i=1}^{10}i = 2 \times 3025 + 5 \times 55
$$

$$
= 6050 + 275 = 6325
$$

**Final Answer:**

$$
\sum_{i=1}^{10}\left(2i^{3} + 5i\right) = 6325
$$
The sum $$ \sum_{i=1}^{10}\left(2i^{3} + 5i\right) $$ equals 6325.

4) \( \sum_{i=1}^{10}\left(2 i^{3}+5 i\right) \)