What is the sum of all odd integers from 35 to 85, inclusive?

To determine the sum of all odd integers from 35 to 85, inclusive, we start by identifying the first and the last odd integers in that range. The first odd integer greater than or equal to 35 is 35 itself, and the last odd integer less than or equal to 85 is 85.

Next, we can see that the odd integer sequence between 35 and 85 is an arithmetic sequence where the first term (a) is 35, the last term (l) is 85, and the common difference (d) is 2.

To find the number of terms (n) in this sequence, we can use the formula for the nth term of an arithmetic sequence:

[

l = a + (n - 1)d

]

Plugging in the values, we have:

[

85 = 35 + (n - 1) \cdot 2

]

Subtracting 35 from both sides gives:

[

50 = (n - 1) \cdot 2

]

Dividing both sides by 2 results in:

[

25 = n - 1

]

So, adding 1 to both sides, we find that there