Question

The sum of 6 consecutive even number is 126. what is the fourth number in the sequence

Answer 1

let's start off by keeping in mind that, if you multiply any integer by 2, regardless of what that integer is, you will always get an EVEN integer, say 13 * 2 = 26, or 18 * 2 = 36 and so on.

let' see a few consecutive even integers

2, 4, 6, 8, 10, 12 , 14..............

notice, to get another one from any of them, we can simply hop back or forwards twice, namely  8 ± 2, gives us 6 and 10.

so let's use for our first even integer say "2a".

that simply means our next even integer can just be 2a + 2

and we add 2 again and so on to get all 6 even integers.

now, we know they add sum up to 126.

$\bf \stackrel{1st}{(2a)}+\stackrel{2nd}{(2a+2)}+\stackrel{3rd}{(2a+4)}+\stackrel{4th}{(2a+6)}+\stackrel{5th}{(2a+8)}+\stackrel{6th}{(2a+10)}~~=~~126 \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ 12a+30=126\implies 12a=96\implies a=\cfrac{96}{12}\implies \boxed{a=8} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \stackrel{4th}{2(8)+6}\implies 22$