(1×2) − 2 = 0 (2×3) − 4 = 2 (3×4) − 6 = 6 (4×5) − 8 = 12 ... $(a\times 12) - b = c$ ... $(d\times e) - f = g$

1. Identify the matching line in the sequence

From the pattern

the two factors are consecutive numbers.

In $(a\times12)-b=c$, since the second factor is $12$, the first factor must be $11$.

So $a=11$.

2. Find the number subtracted and calculate the result

The number subtracted is twice the first factor:
$2,4,6,8,\ldots$
So when $a=11$,
$b=2\times11=22.$
Then
$c=(11\times12)-22=132-22=110.$

1. Identify the relationship between the terms in each equation

From the sequence:
$(1\times2)-2=0$
$(2\times3)-4=2$
$(3\times4)-6=6$
Each line has the form:
$n(n+1)-2n$
So for $(d\times e)-f=g,$ we have $e=d+1$ and $f=2d$.

2. Express $g$ in terms of $d$ and simplify

$g=d(d+1)-2d$
$g=d^2+d-2d$
$g=d^2-d$

1. Use the general form of the result and compare its parity with 279

From part (b), the result is
$g=d^2-d=d(d-1).$
Since $d$ and $d-1$ are consecutive integers, one of them is even, so $d(d-1)$ is always even.

But $279$ is odd, so $279$ cannot be a result in this sequence.