Question

The number 75 can be expressed as the sum of two or more consecutive positive integers in five different ways. One such sequence begins with 13: 13 + 14 + 15 +16 +17= 75. With what number does each of the four sequences begin?

Answer 1

X, x+1, x+2, x+3, ... - consecutive integers

75 as the sum of two consecutive integers:
$x+x+1=75 \\ 2x+1=75 \\ 2x=75-1 \\ 2x=74 \\ x=\frac{74}{2} \\ x=37 \\ \\ \boxed{75=37+38}$

75 as the sum of three consecutive integers:
$x+x+1+x+2=75 \\ 3x+3=75 \\ 3x=75-3 \\ 3x=72 \\ x=\frac{72}{3} \\ x=24 \\ \\ \boxed{75=24+25+26}$

75 as the sum of six consecutive integers:
$x+x+1+x+2+x+3+x+4+x+5=75 \\ 6x+15=75 \\ 6x=75-15 \\ 6x=60 \\ x=\frac{60}{6} \\ x=10 \\ \\ \boxed{75=10+11+12+13+14+15}$

75 as the sum of ten consecutive integers:
$x+x+1+x+2+x+3+...+x+7+x+8+x+9= 75 \\ 10x+45=75 \\ 10x=75-45 \\ 10x=30 \\ x=\frac{30}{10} \\ x=3 \\ \\ \boxed{75=3+4+5+6+7+8+9+10+11+12}$

The four sequences begin with: 3, 10, 24, 37.