Advanced Math/Algebra Question
- Determine the sum of three consecutive natural numbers if the 1st one is n+5.
- The answer that was given in the corrections was "3n+18"
- Determine the sum of three consecutive odd numbers if the 2nd one is "2n+11"
- The answer that was given in the corrections was "6n+33"
- My math teacher gave the corrections to us by email and they had no explanation. I am having difficulty in finding out how to get those two answers.
The key to these problems is to understand what is meant by consecutive numbers and how the letter n is used. Natural numbers are the easiest and most familiar numbers to deal with. They are also known as the "counting" numbers because consecutive numbers follow each other just like counting. For a given natural number, that can be represented by the symbol n or an expression involving n, the next number in the counting sequence is obtained by just adding 1. Examples:
- the number consecutive to 3 is 4
- the number consecutive to n is n+1
- in n = 3, then n +1 is 4
- the number consecutive to 2n+5 is (2n+5) + 1 = 2n+6.
Note that we can go backwards too; if n is a consecutive number, the number before it in the sequence is n-1
- eg., if 4 is the consecutive number, then (4-1) = 3 is the number before it.
Three consecutive numbers follow the same pattern, i.e., 3, 4, 5.
The sum of three consecutive numbers is just (n) + (n+1) + (n+2) = n + n + n + 1+ 2 = 3n + 3.
Turning to your examples, if the first number in a sequence of consecutive numbers is n+5 then the next 2 numbers are (n+5) + 1 = (n+6) and (n+5) + 2 = (n+6) + 1 = (n+7), or, written out in a row
(n+5), (n+6), (n+7).
The sum is just
(n+5) + (n+6) + (n+7) = n+n+n+5+6+7 = 3n+18.
If a natural number 2n+11 is the second in a sequence, then the sequence will be
(2n+11) - 1, (2n+11), (2n+11) + 1, which is the same as (2n+10), (2n+11), (2n+12). The sum is then
(2n+10) + (2n+11) + (2n+12) = 3(2n) + 10+11+12 = 6n +33.
Pretty straightforward, really. Keep up the good work.