The problem is to evaluate the sum $\sum_{i=6}^{20} 2i - 5$.

1. Problem Description

The problem is to evaluate the sum $\sum_{i=6}^{20} 2i - 5$.

2. Solution Steps

First, we can split the summation:

$\sum_{i=6}^{20} (2i - 5) = \sum_{i=6}^{20} 2i - \sum_{i=6}^{20} 5$

Then, we can pull out the constant 2 from the first term:

$= 2\sum_{i=6}^{20} i - \sum_{i=6}^{20} 5$

Now, we need to evaluate $\sum_{i=6}^{20} i$. This is the sum of integers from 6 to

2

0. We can use the formula for the sum of the first $n$ integers, which is given by:

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

Then, we can calculate the sum from 1 to 20 and subtract the sum from 1 to

5. $\sum_{i=6}^{20} i = \sum_{i=1}^{20} i - \sum_{i=1}^{5} i = \frac{20(20+1)}{2} - \frac{5(5+1)}{2} = \frac{20(21)}{2} - \frac{5(6)}{2} = \frac{420}{2} - \frac{30}{2} = 210 - 15 = 195$

Now, we need to evaluate $\sum_{i=6}^{20} 5$. This is simply adding the constant 5 for each value of $i$ from 6 to

2

0. The number of terms in the summation is $20 - 6 + 1 = 15$.

$\sum_{i=6}^{20} 5 = 5(15) = 75$

Substituting these values back into the expression:

$2\sum_{i=6}^{20} i - \sum_{i=6}^{20} 5 = 2(195) - 75 = 390 - 75 = 315$

3. Final Answer

315