consecutive natural numbers and avererage 8, 9, 10, 11, 12
Answers
Answer:
Age 11 to 14
Swaathi, from Garden International School, started by listing the numbers up to 15 and trying to represent them as sums of consecutive numbers:
2
3 = 1+2
4
5 = 2+3
6 = 1+2+3
7 = 3+4
8
9 = 4+5 = 2+3+4
10 = 1+2+3+4
11 = 5+6
12 = 3+4+5
13 = 6+7
14 = 2+3+4+5
15 = 7+8 = 4+5+6 = 1+2+3+4+5
We can't write every number as a sum of consecutive numbers - for example, 2, 4 and 8 can't be written as sums of consecutive numbers. In the above, 9 and 15 were the only numbers that I could find that could be written in more than one way.
Many people spotted the pattern that all odd numbers (except 1) could be written as the sum of two consecutive numbers. For example, Matilda and Tamaris wrote:
If you add two consecutive numbers together, the sum is an odd number, e.g.
1+2=3
2+3=5
3+4=7
4+5=9
5+6=11
6+7=13
and so on...
Well done to pupils from Kenmont Primary School who noticed this, and explained that an Odd plus an Even is always Odd.
Some spotted a similar pattern for multiples of 3. Julia and Lizzie said:
If you add any 3 consecutive numbers together it will always equal a multiple of 3, e.g.
1+2+3=6
2+3+4=9
3+4+5=12
4+5+6=15
5+6+7=18
Continuing with the patterns, the Lumen Christi grade 5/6 maths extension program team sent us:
We discovered that the sum of four consecutive numbers gave us the number sequence 10, 14, 18, 22, 26, 30, and so on. They were all even numbers that had an odd number as half of its total.
1+2+3+4=10
2+3+4+5=14
3+4+5+6=18...
Heather from Wallington High School for Girls explained this pattern:
10 - 1+2+3+4
14 - 2+3+4+5
18 - 3+4+5+6
22 - 4+5+6+7
In all the columns, each place adds 1 each time, so in total you add 4 each time.
Ruby said:
Numbers which are multiples of 5, starting with 15, are sums of 5 consecutive numbers:
1+2+3+4+5=15
2+3+4+5+6=20
3+4+5+6+7=25...
Fergus and Sami noticed a similar pattern:
If you allow negative numbers, you can find a sum for any multiple of 7 easily. Each time you add one number either side of the sum, your sum increases by 7, e.g.
3+4=7
2+3+4+5=14
1+2+3+4+5+6=21
0+1+2+3+4+5+6+7=28
-1+0+1+2+3+4+5+6+7+8=35...
Great! (There's a way to make this pattern work even without using negative numbers - can you spot it?) Why are all these patterns arising?
Becky spotted a different type of pattern:
We found out that powers of 2 (2, 4, 8, 16...) can never be made by adding together consecutive numbers together.
Interesting! I wonder why?
The Lumen Christi team give a way of constructing lots of multiples of odd numbers:
We worked out that if you divide a multiple of 3 by 3, and call the answer n, then your original number is the sum of (n-1), n and (n+1).
Then we discovered that the multiples of 5 can be written as 5 consecutive numbers. It's the same as the rule for 3 consecutive numbers. Take a number and divide it by 5, call it n, and then your number is the sum of (n-2), (n-1), n, (n+1) and (n+2).
We then made a conjecture that since it is true for 3 and 5, it would also work for 7, 9 and any other odd number. We tested it, and it worked. For example, 63 is a multiple of 7 and 9:
7 numbers: 6+7+8+9+10+11+12=63
(63/7 = 9)
9 numbers: 3+4+5+6+7+8+9+10+11=63
(63/9 = 7)
Step-by-step explanation:
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