To solve the problem of finding the sum of all odd integers from 30 to 80, let's proceed step-by-step.

1. Identify the range of odd numbers: The odd numbers between 30 and 80 are: $31, 33, 35, \ldots, 79$

2. Form an arithmetic sequence: This sequence is arithmetic with:

   • First term $a = 31$
   • Last term $l = 79$
   • Common difference $d = 2$

3. Find the number of terms ($n$): The formula for the $n$-th term of an arithmetic sequence is: $a_n = a + (n - 1) \cdot d$ Setting $a_n = 79$ (the last term), we can solve for $n$: $79 = 31 + (n - 1) \cdot 2$ $79 - 31 = 2(n - 1)$ $48 = 2(n - 1)$ $24 = n - 1$ $n = 25$ So, there are 25 terms in this sequence.

4. Use the sum formula for an arithmetic sequence: The sum $S$ of an arithmetic sequence is given by: $S = \frac{n}{2} \cdot (a + l)$ Substituting the values we have: $S = \frac{25}{2} \cdot (31 + 79)$ $S = \frac{25}{2} \cdot 110$ $S = 25 \cdot 55$ $S = 1375$

Therefore, the sum of the odd integers from 30 to 80 is 1375. The correct answer is:

B. 1375

Would you like further details on any step?

Here are some additional questions for exploration:

1. How would the solution change if we needed the sum of even integers instead?
2. What is the general formula to find the sum of an arithmetic sequence?
3. How does the sum change if the range were from 20 to 90 instead?
4. Can you verify this sum by adding each term manually?
5. What if we only wanted the count of odd numbers in this range without summing them?

Tip: For sums of consecutive numbers, using the arithmetic sequence formula simplifies calculations significantly!