Question

The sum of 5 consecutive even numbers is 180 what is the first number in the sequence

Answer 1

First, a even number in general could be written as  2k for any k ( k is any integer number).

Let x_1, x_2, x_3, x_4, x_5 the 5 consecutive even numbers, then,

x_1 + x_2 + x_3 + x_4 + x_5 = 180. But for the first commentary, any of the x_i's are even numbers, so, are of the form 2k_i, but, they are consecutive, think about the sequence: 2,4,6,8,10, we could written as 2(1) + 2 (2) + 2(3) + 2(4) + 2(5), the numbers between (), are the k's and are consecutive, and we could write consecutive numbers as x, x+1, x+1 +1 = x+2, (x+2)+1, etc.

So, for the previous commentary, we rewrite

x_1 + x_2 + x_3 + x_4 + x_5 = 180

as

2k + 2(k+1) + 2(k+2) + 2(k+3) + 2(k+4)  = 180

is equivalent to (factorizing 2)

2 [k + k+1 + k+2 +k+3 +k+4 ] = 180

(divide both sides by 2)

[k + k+1 + k+2 +k+3 +k+4 ] = 90

(simplifyng the expression)

5k+10 = 90

(substracting 10 in both sides)

5k = 80

(divide both sides by 5)

k = 16

then, the first number is x_1 = 2k = 2(16) = 32

Now, we can check, the corresponding sequence was

32 + 34 +36 +38 +40 = 180.