To solve the problem of finding three consecutive odd numbers whose sum is 75, follow these steps:
1. Define the variables:
Let the three consecutive odd numbers be [tex]\( n \)[/tex], [tex]\( n+2 \)[/tex], and [tex]\( n+4 \)[/tex].
2. Set up the equation:
According to the problem, the sum of these three consecutive odd numbers is 75. Therefore, we can write the equation as:
[tex]\[
n + (n + 2) + (n + 4) = 75
\][/tex]
3. Simplify the equation:
Combine like terms on the left side:
[tex]\[
n + n + 2 + n + 4 = 75
\][/tex]
This simplifies to:
[tex]\[
3n + 6 = 75
\][/tex]
4. Solve for [tex]\( n \)[/tex]:
To isolate [tex]\( n \)[/tex], first subtract 6 from both sides of the equation:
[tex]\[
3n + 6 - 6 = 75 - 6
\][/tex]
Which simplifies to:
[tex]\[
3n = 69
\][/tex]
Next, divide both sides by 3:
[tex]\[
\frac{3n}{3} = \frac{69}{3}
\][/tex]
This gives:
[tex]\[
n = 23
\][/tex]
5. Find the three consecutive odd numbers:
Now that we have the value of [tex]\( n \)[/tex], we can determine the three consecutive odd numbers:
[tex]\[
n = 23
\][/tex]
[tex]\[
n + 2 = 25
\][/tex]
[tex]\[
n + 4 = 27
\][/tex]
Therefore, the three consecutive odd numbers whose sum is 75 are [tex]\( 23 \)[/tex], [tex]\( 25 \)[/tex], and [tex]\( 27 \)[/tex].
