patterns for math 1 4 9 16 25 36?

Question

Okay, I know the pattern for this is just the numbers (1, 2, 3, ect) squared.

My question is, why is there also the pattern of adding odd numbers starting at 3?

For example 1+3=4 4+5=9 9+7=16 16+9=25 11+13=36

Okay you're right... ahhh 25+11=36... yeah, sorry

Puzzling, you ROCK! I totally understood your explaination of just thinking about the squares visually, and you will see how it keepsadding 3, and, 5, and 7,etc.

However, I don't understand the second part of you explaination:

"Let's take the function:
f(x) = x².

Now figure out what would happen if you took f(x + 1) - f(x)? The answer would be (x+1)² - x². Expand that out and you get:
(x² + 2x + 1) - x² = 2x + 1"

However, the part I didn't get was why did you do this: f(x + 1) - f(x)?

I understand how the fuction is x²
And I understand why you would want to do f(x+1), because each number in my sequence is 1 more than the next, and then squared. By why are you taking away f(x) from f(x+1)

HAHA I am soooo sloow, NEVER MIND! I thought about it some more, and I understand now completely why you would do the f(x+1) - f(x). So everyone ignore my last detail. LOL. I'm crazy.

Ok, I honestly dont think anyone is still reading my details. But in case you are.. can any explain Yes-I'm-French's answer? Like what on earth does a1 mean? does "a" mean answer? so it's the answer to 1?

Answer 1

1 + 3 + 5 + ..... + (2n-1) = n^2
you are adding the first n terms of an arithmetic sequence with
first term a1=1 , common difference is 2 so a2 = a1 + 2

the general answer for the n-term arithmetic sequence sum is Sn = (n/2) ( a1 + an)
or an = a1 + (n - 1) d where d is 2 ( common difference)
so an = 1 + (n - 1)2 = 2n - 1
then Sn = (n/2) ( 1 + 2n -1)
Sn = (n/2) (2n) = (2n²)/2 = n²
so indeed the sum of the first 2 terms (1+3=4) is 2²
sum of the first 3 1+3+5=9=3²

edit
the pattern you found is a sequence of numbers
actually its an arithmetic sequence because you jump from one number to next one by adding always the same number (here it's 2 aka common difference)
a1 is just the name of the first number, a2 is the second, a3 the third...
so you see that a2= a1 + 2 (you jump from une number to the next one adding 2 your common difference)
same for a3 = a2 + 2 and so on
let's say i want to calulate a234, i guess that will be pretty long adding 2 everytime
so mathematicians invented formulas to calculate a asomething without knowing the previous number
an = a1 + (n - 1) d
an is any term of the sequence like the 234th one so n = 234 for that exemple, a1 is still the first term of your sequence ( 1 for your's) and, d the common difference ( 2 for your's)
so a234 = 1 + ( 234 - 1) 2 = 467
and since the sequences are really usefull to mathematics they made more formulas, like one to calculate the sum of the n firsts terms of a sequence Sn = (n/2) ( a1 + an)
for the 234 first terms the sum would be
S234 = ( 234/2) (1+467)= 54756 and since i showed it a bit earlier S234 is 234² for your specific sequence 234²=54756

hope it's clearer, if you need more explanations you know where to ask =)
would be easier with some real math fonts =)
sequences/ series/ recursion dunno really how you call em in the USA are an amazing field of mathematics used in a LOOOOOT of everyday applications, and have been studied by almost all of the greatest mathematicians

Answer 2

Imagine you had a set of items. Put 1 put down. Okay, that represents 1 or 1².

Now what are you going to do to make it into a square of 2 x 2? You are going to add 1 above and to the right, then 1 more on the diagonal. 2(1) + 1 is an odd number, namely 3. So you are adding 3 and you get to 4 (2²).

Now how are you going to get to a square of 3 x 3? You are going to add 2 on the top and right and one more on the diagonal. That is another odd number 2(2) + 1 = 5 which make 9 (3²).

You can continue the pattern and each time it is an increasing odd number of the form 2n + 1.

However, you can also prove it algebraically rather than empircally.

Let's take the function:
f(x) = x².

Now figure out what would happen if you took f(x + 1) - f(x)? The answer would be (x+1)² - x². Expand that out and you get:
(x² + 2x + 1) - x² = 2x + 1

So the difference between consecutive squares is 2x + 1 --> and if you think about the sequence that forms... it is odd numbers.

It's not just coincidence, it's a fact. (btw, I think you meant to have 25+11 = 36 and 36+13 = 49 in your example, but I got what you meant.)

Answer 3

The difference between the sets increase by the odd number.

So...difference between 1 and 4 is 3. Difference between 4 and 9 is five.

Next number in pattern is 49.

Answer 4

1. Integer values follow an odd, even, odd, even pattern.
2. Odd numbers sqaured are always odd numbers.
3. Even numbers squared are always even numbers.

Given 1, 2 and 3:
4. Squares of integers follow an odd, even, odd, even pattern.
5. The difference of any odd number from and even number is always odd.

Answer 5

11+13=24, not 36

Answer 6

well the patern is the perfect squares and the 3 thing isn't true, os i guess thatanswers your questions

Answer 7

You asked: "why is there also the pattern".

The answer is, "because someone made up that pattern".