The problem is to evaluate the summation $\sum_{i=4}^{30} 4i$.

1. Problem Description

The problem is to evaluate the summation $\sum_{i=4}^{30} 4i$.

2. Solution Steps

We need to find the sum of the terms $4i$ as $i$ ranges from 4 to

3

0. We can factor out the constant 4 from the summation:

$\sum_{i=4}^{30} 4i = 4 \sum_{i=4}^{30} i$

The sum of the first $n$ integers is given by the formula:

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

We can rewrite the summation from $i=4$ to $30$ as:

$\sum_{i=4}^{30} i = \sum_{i=1}^{30} i - \sum_{i=1}^{3} i$

Using the formula for the sum of the first $n$ integers, we have:

$\sum_{i=1}^{30} i = \frac{30(30+1)}{2} = \frac{30(31)}{2} = 15(31) = 465$

$\sum_{i=1}^{3} i = \frac{3(3+1)}{2} = \frac{3(4)}{2} = 6$

Therefore,

$\sum_{i=4}^{30} i = 465 - 6 = 459$

Now, we multiply this by 4:

$4 \sum_{i=4}^{30} i = 4(459) = 1836$

3. Final Answer

1836