To find the sum of the odd numbers between 16 and 80, we can use the formula for the sum of an arithmetic series. The odd numbers between 16 and 80 form an arithmetic sequence.

Step-by-step process:

1. List the first and last odd numbers:

   • The first odd number after 16 is 17.
   • The last odd number before 80 is 79.

2. Find the number of terms: The common difference $d$ of odd numbers is 2. The $n$-th term of an arithmetic sequence can be given by: $a_n = a_1 + (n-1) \cdot d$ Here, $a_n = 79$, $a_1 = 17$, and $d = 2$. Plugging in these values: $79 = 17 + (n-1) \cdot 2$ Solving for $n$: $79 - 17 = (n-1) \cdot 2 \quad \Rightarrow \quad 62 = (n-1) \cdot 2 \quad \Rightarrow \quad n-1 = 31 \quad \Rightarrow \quad n = 32$ So, there are 32 terms.

3. Sum of an arithmetic series formula: The sum of an arithmetic sequence is given by: $S_n = \frac{n}{2} \cdot (a_1 + a_n)$ Substituting the known values: $S_{32} = \frac{32}{2} \cdot (17 + 79) = 16 \cdot 96 = 1536$

Thus, the sum of the odd numbers between 16 and 80 is 1536.

Would you like further details or explanations?

Here are 5 related questions:

1. What is the sum of the even numbers between 16 and 80?
2. How can you derive the formula for the sum of an arithmetic sequence?
3. What is the sum of odd numbers between 1 and 100?
4. What is the general approach for summing any arithmetic sequence?
5. How would the method change if we were summing all numbers (not just odd)?

Tip: When solving arithmetic sequence problems, identifying the first and last terms and the common difference is key to using the correct formula.