x, x+1, x+2, x+3, ... - consecutive integers
75 as the sum of two consecutive integers:
[tex]x+x+1=75 \\
2x+1=75 \\ 2x=75-1 \\
2x=74 \\
x=\frac{74}{2} \\
x=37 \\ \\ \boxed{75=37+38}[/tex]
75 as the sum of three consecutive integers:
[tex]x+x+1+x+2=75 \\
3x+3=75 \\
3x=75-3 \\
3x=72 \\
x=\frac{72}{3} \\
x=24 \\ \\ \boxed{75=24+25+26}[/tex]
75 as the sum of six consecutive integers:
[tex]x+x+1+x+2+x+3+x+4+x+5=75 \\
6x+15=75 \\
6x=75-15 \\
6x=60 \\
x=\frac{60}{6} \\
x=10 \\ \\
\boxed{75=10+11+12+13+14+15}[/tex]
75 as the sum of ten consecutive integers:
[tex]x+x+1+x+2+x+3+...+x+7+x+8+x+9= 75 \\
10x+45=75 \\
10x=75-45 \\
10x=30 \\
x=\frac{30}{10} \\
x=3 \\ \\
\boxed{75=3+4+5+6+7+8+9+10+11+12}[/tex]
The four sequences begin with: 3, 10, 24, 37.
