# Consecutive Odd Numbers?

When doing some exercises on Khan Academy, the following question came up: "The sum of 3 consecutive odd numbers is 69. What is the third number in this sequence?" My answer was 27 because I thought -Odd- Numbers to show up as = 1, 3, 5, 7, etc. Apparently, it consists of x, 2, 4, 6, etc. Can someone explain this to me?

• English is not my first or second language in case I'm missing something..

• Your definition of odd is correct; $0, 2, 4, 6, ...$ are even numbers. 27 is wrong because 23 + 25 + 27 is 75, not 69. – user296602 Aug 10 '17 at 3:22
• Uhm thanks but how did you get to 75? The answer is 25. – Eli S. Aug 10 '17 at 3:24
• Let $x$ be an odd number, then $x+(x+2)+(x+4)=69$ implies $x=21$ so the third number is $25$ – Chinnapparaj R Aug 10 '17 at 3:24
• @EliS. My point is that you can tell your answer is wrong because 23 + 25 + 27 = 75, as you can add by hand. – user296602 Aug 10 '17 at 3:25
• Oh yeah, thanks user, I know it's wrong but I just didn't get why they used even numbers instead of odd numbers. Thanks Chinz, I get it now. Since x is 21 = Odd number, +2 is 23 + 4 is 25 which are all odd. Thanks! – Eli S. Aug 10 '17 at 3:27

Consecutive odd numbers differs $2$.

Now suppose $a$ is a odd numbers, then $a-2, a+2$ are also odd numbers.

Now it is given that $$a-2+a+a+2=69\\\Rightarrow 3a=69\\\Rightarrow a=23$$

Then the third number in this sequence is $a+2=25$ (Ans.)

Odd numbers refer to the number that are not divisible by 2. E.g. 1, 3, 5, ...

As for the question, we need to assume 3 consecutive odd numbers: x, x+2 and x+4. (the difference between 2 consecutive-odds is 2)

$$x+x+2+x+4=69$$ $$3x+6=69$$ $$3x= 69-6=63$$ $$x={63\over 3} = 21$$

Hence, the third number is 21+4= 25.

• Thanks Kummar, already solved it in the comments too. :) – Eli S. Aug 10 '17 at 3:45

Using arithmetic progression formula

$$S_n=\frac{n}{2}(a+l)$$

where a is the first term and l is the last term.

$$S_n=\frac{n}{2}(2a+(n-1)d)$$

$$S_3=69$$

The difference between two odd numbers is two.

odd number sequences

$$1,3,5,7,9...$$

It is obvious that the sequence is arithmetic and have a common difference,

$$3-1=2=5-3$$

$$d=2$$

$$S_n=\frac{n}{2}(2a+(n-1)d)$$

where d is the common difference and a is the first term.

$$69=\frac{3}{2}(2a+(3-1)2)$$

$$a=21$$

By

$$S_n=\frac{n}{2}(a+l)$$

$$69=\frac{3}{2}(21+l)$$

$$l=25$$

• This seems unnecessarily convoluted. Why not just observe that if the first odd number is $N$, then the other two are $N+2$ and $N+4$, and so the statement is that their sum $3N+6$ equals $69$, i.e., $3N+6=69$? This leads immediately to $N=21$ and so to $N+4=25$. – MPW Aug 10 '17 at 3:40