The sum of even numbers between 1 and 31:

Question

Accepted Answer

To determine the sum of even numbers between 1 and 31, we first identify the even numbers in this range. The even numbers between 1 and 31 are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, and 30.

Now, we calculate the sum of these even numbers:

$$ 
2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 
$$

To simplify the calculation, we can pair the numbers from the beginning and end of the sequence:

- (2 + 30) = 32
- (4 + 28) = 32
- (6 + 26) = 32
- (8 + 24) = 32
- (10 + 22) = 32
- (12 + 20) = 32
- (14 + 18) = 32

The number 16 is left unpaired, as there are an odd number of terms.

Now, we have 7 pairs of 32, plus the unpaired 16:

$$ 
7 \times 32 + 16 = 224 + 16 = 240 
$$

Thus, the sum of even numbers between 1 and 31 is 240, which corresponds to Option 3.

**Explanation for Incorrect Options:**

1. **Option 1 (16):** This option is incorrect because 16 is just one of the even numbers in the sequence, not the sum of all even numbers between 1 and 31.

2. **Option 2 (128):** This option is incorrect because the sum of the even numbers is much larger than 128. As calculated, the correct sum is 240.

3. **Option 4 (512):** This option is incorrect because 512 is significantly larger than the actual sum of 240. The sum of the even numbers does not reach this value.

Therefore, based on the calculations and logical reasoning, Option 3 is indeed the correct answer.