To solve this problem:

Given:

$\sum_{n=1}^{20} (3 + 5n)$ We are asked:

1. How much the sum would change if $n$ starts at 0 and ends at 19.

Step 1: Calculate the original sum from $n = 1$ to $n = 20$

The summation expression is $3 + 5n$. To compute the sum:

1. Split the terms: $\sum_{n=1}^{20} (3 + 5n) = \sum_{n=1}^{20} 3 + \sum_{n=1}^{20} 5n$

2. Calculate each part:

   • First part: $\sum_{n=1}^{20} 3$
     Since 3 is a constant added 20 times: $\sum_{n=1}^{20} 3 = 3 \cdot 20 = 60$

   • Second part: $\sum_{n=1}^{20} 5n$
     Factor out the 5: $\sum_{n=1}^{20} 5n = 5 \cdot \sum_{n=1}^{20} n$ Use the formula for the sum of the first $N$ natural numbers: $\sum_{n=1}^{20} n = \frac{20(20+1)}{2} = \frac{20 \cdot 21}{2} = 210$ Thus: $\sum_{n=1}^{20} 5n = 5 \cdot 210 = 1050$

3. Combine the results: $\sum_{n=1}^{20} (3 + 5n) = 60 + 1050 = 1110$

Step 2: Recalculate the sum from $n = 0$ to $n = 19$

Now, $n$ starts at 0 and ends at 19. The same expression applies:

1. Split the terms: $\sum_{n=0}^{19} (3 + 5n) = \sum_{n=0}^{19} 3 + \sum_{n=0}^{19} 5n$

2. Calculate each part:

   • First part: $\sum_{n=0}^{19} 3$
     Since 3 is added 20 times (including $n = 0$): $\sum_{n=0}^{19} 3 = 3 \cdot 20 = 60$

   • Second part: $\sum_{n=0}^{19} 5n$
     Factor out the 5: $\sum_{n=0}^{19} 5n = 5 \cdot \sum_{n=0}^{19} n$ Use the formula for the sum of the first $N$ natural numbers, where $N = 19$: $\sum_{n=0}^{19} n = \frac{19(19+1)}{2} = \frac{19 \cdot 20}{2} = 190$ Thus: $\sum_{n=0}^{19} 5n = 5 \cdot 190 = 950$

3. Combine the results: $\sum_{n=0}^{19} (3 + 5n) = 60 + 950 = 1010$

Step 3: Determine the change in the sum

The original sum (from $n = 1$ to $n = 20$) is 1110.
The modified sum (from $n = 0$ to $n = 19$) is 1010.

The change in the sum is: $1110 - 1010 = 100$

Final Answer:

• The sum decreases by 100.

Do you want further clarification or a detailed breakdown of any part?

Here are 5 related questions you might find useful:

1. What is the formula for summing an arithmetic series?
2. How do you calculate the sum of the first $N$ natural numbers?
3. How does changing the bounds of a summation affect its result?
4. What happens when constants are added inside summation notation?
5. How can you simplify a summation involving both constants and linear terms?

Tip: Always split summations into simpler parts when dealing with sums of constants and variables!