The sum of 6 consecutive even number is 126. what is the fourth number in the sequence
1 answer:
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.
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.
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which gives the critical point
We can confirm that this is a minimum by checking the Hessian matrix of:
is positive definite (we see its determinant and the determinants of its leading principal minors are positive), which indicates that there is a minimum at this critical point.
At this point, we get a distance from (0, 2, 4) of
but since
The Lagrangian is
with partial derivatives (set equal to 0)
Solve for
which gives the critical point
We can confirm that this is a minimum by checking the Hessian matrix of
At this point, we get a distance from (0, 2, 4) of