Question 1:
A) n=2 s=2+4=6 / n=5 s=2+4+6+8+10=30
s=2+4+6=12
B) n=1=1 n=2=3 n=3=6 n=4=10
n=1=2 n=2=6 n=3=12 n=4=20 (if you take the sum of the first n integars, and then you take the sum of the first n even integars, you will notice that they double)
C) It makes sense because when you are adding the even numbers, you actually are doubling the sum. All the even numbers are doubled and can be divided by two so this is what happens when you do this.
Question 2:
A) n=1 n=1
n=2 n=1+3=4
n=3 n=1+3+5=9
n=4 n=1+3+5+7=16
n=5 n=1+3+5+7+9=25
n=6 n=1+3+5+7+9+11=36
n=7 n=1+3+5+7+9+11+13=49
n=8 n=1+3+5+7+9+11+13+15=63
B) If you didn't want to do all this adding, all you would have to do it square your number. Each number is the square root of the sums.
C) This holds true for all n because if you kept going with this, you will see that it will continue to square out througout all numbers. Numbers are made to work out the way they do.
Your post contains a few of the important observations that I wanted people to figure out. Mainly that for the evens, it's the same as last week but doubled (and from that if you look at my response post that gives you the correct formula for last week, you can get the generic formula I asked for.)
ReplyDeleteYou also noted that for the odds, the formula is just n^2.
The only thing I wished to see more out of was question 2C, but nonetheless, good job!