5. The sum of the first 20 natural numbers is $190$, what is the average number in this sequence? How would the answer change if the sum of the entire consecutive integers was $990$?

Expert intro: For consecutive integers, the average is the total sum divided by the number of terms. If you know the sum and the count, the mean is straightforward to compute.

Detailed walkthrough

Part 1: Average of the first 20 natural numbers

The first 20 natural numbers are

$1,2,3,\dots,20$

Their sum is $190$, so the average is

$\frac{190}{20} = 9.5$

Answer for the first part

$\boxed{9.5}$

Part 2: If the sum of the consecutive integers is $990$

If the sequence is $1+2+3+\cdots+n = 990$, then

$\frac{n(n+1)}{2} = 990$

$n(n+1) = 1980$

Checking factors gives $n=44$, because

$\frac{44\cdot45}{2} = 990$

So the average of the sequence $1,2,\dots,44$ is

$\frac{990}{44} = 22.5$

Answer for the second part

$\boxed{22.5}$

💡 Pitfall guide

Do not confuse the average of the first 20 natural numbers with the average of the first 20 terms in an arithmetic sequence centered around the middle value. Here the average is simply sum divided by count.

🔄 Real-world variant

If the sequence were not consecutive integers, the average would not be determined just by the sum. You would also need the number of terms, and possibly the first and last terms.

🔍 Related terms

mean, consecutive integers, sum of natural numbers