For adding two consecutive square numbers and then subtracting 1,
We find a sequence of 4,12,24,40,60 so each number is increasing consecutively with an additional 4 starting with 4 and 12 having a difference of 8,then 12,and then 16. They are all multiples of 4
Let us say a is a number so a^2 is a square number so when we add a consecutive square number we get a^2+(a+1)^2-1= 2a^2+2a= 2a(a+1). If 2a were odd then a+1 would be even. They would both have at least have a factor of 2 so they both would be multiples of 4.
For squaring any odd number, then subtract 1
3^2-1=8 , 5^2-1=24, 7^2-1=48 , 9^2-1=80
We notice that all these numbers are multiples of 8. Further if we divide theese numbers of 8 we get another sequence with a consecutive addition of 1 to each difference. So 1,3,6,10 . Difference of 2 then 3 then 4.
Let us say a is an even number so a+1 is odd. Squaring and subtracting 1,
(a+1)^2-1=a^2+2a=a(a+2)
Since a is even a can only be values of 2,4,6 etc. This means that one side is at least a multiple of 2 and the other a multiple of 4.
For multiplying two consecutive odd numbers and then adding 1,
1*3+1=2^2, 3*5+1=4^2, 5*7+1=6^2 , 7*9+1=8^2. They are all even numbers squared.
Now again saying a is an odd number and (a+1) is an even. Multiplying 2 odd consecutives we get:
(a+1)(a+3)+1=a^2+2a+4=(a+2)^2 and since a is even we will always have an even square number.
For multiplying two consecutive even numbers and then adding 1,
2*4+1=3^2, 4*6+1=5^2 ,6*8+1=7^2. These are all odd square numbers.
By same argument say a is an odd number and (a+1) is an even number. Multiplying two consecutive even numbers:
(a+1)(a+3)+1=a^2+2a+4=(a+2)^2 . Since a is an odd number there will always be an odd square number.
Here are a couple I have created:
If you have 4 consecutive square numbers, say 9,16,25,36 then add up the 1st and 4th numbers and subtract from it the sum of the 2nd and 3rd numbers. What do you notice?
If you add up two square numbers that are apart by one e.g 1 and 9, 4 and 16, then find the sum of the 2 numbers.What do you notice?