Question

Look for a pattern in the first three equations. Then, fill in the missing numbers in the rest of the series of equations. 1= 1 · 2⁄2 1 + 2 = 2 · 3⁄2 1 + 2 + 3 = 3 · 4⁄2 1 + 2 + 3 + ? = ? · ?⁄? 1 + 2 + 3 + ? + ? = ? · ?⁄?

Answer 1

1=1 times 2/2 1 + 2=2 times 3/2 1 +2+3=3 times 4/2 1+2+3+4=4 times 5/2 1+2+3+4+5=5 times 6/2

Answer 2

`1+2+3+4+=4* (5)/(2)\\ 1+2+3+4+5=5 * (6)/(2)`

Explanation:

The given equation are

`1=1* (2)/(2) \\1+2=2* (3)/(2)\\ 1+2+3=3* (4)/(2)\\ 1+2+3+?=?* (?)/(?)\\ 1+2+3+?+?=?* (?)/(?)`

As you can observe, the left side of equation has a sum which increases by one. The right side of the equation has a product between a number and a fraction, the number increases 1 unit, and the numerator of the fraction also increases 1 unit, but the denominator remains the same.

Applying this patterns, the third and fourth equation are

`1+2+3+4+=4* (5)/(2)\\ 1+2+3+4+5=5 * (6)/(2)`

Therefore, the two last equation completed are

`1+2+3+4+=4* (5)/(2)\\ 1+2+3+4+5=5 * (6)/(2)`