The sum of 6 consecutive odd numbers is 20. What is the fourth number in this sequence?

1 Answer
Jan 2, 2017

There is no such sequence of 6 consecutive odd numbers.

Explanation:

Denote the fourth number by n.

Then the six numbers are:

n-6, n-4, n-2, color(blue)(n), n+2, n+4

and we have:

20 = (n-6)+(n-4)+(n-2)+n+(n+2)+(n+4)

color(white)(20) = (n-6)+5n

color(white)(20) = 6n-6

Add 6 to both ends to get:

26 = 6n

Divide both sides by 6 and transpose to find:

n = 26/6 = 13/3

Hmmm. That is not an integer, let alone an odd integer.

So there is no suitable sequence of 6 consecutive odd integers.

color(white)()
What are the possible sums of a sequence of 6 consecutive odd numbers?

Let the average of the numbers be the even number 2k where k is an integer.

Then the six consectuvie odd numbers are:

2k-5, 2k-3, 2k-1, 2k+1, 2k+3, 2k+5

Their sum is:

(2k-5)+(2k-3)+(2k-1)+(2k+1)+(2k+3)+(2k+5) = 12k

So any multiple of 12 is a possible sum.

Perhaps the sum in the question should have been 120 rather than 20. Then the fourth number would be 21.