Conjecture..I dont know how to word it

[K.ourt]

1) Make a conjecture about the next 3 terms in the sequence 1,5,2,8,3,11,... Explain your reasoning.

Ok, so I know it goes +4,-3,
+6,-5,+8....So, is it something like: add to the number x-1, then x+3 to the sum that you get, and so on and so forth? Because the numbers that are added/subtracted decrease by one, then increase by three. Its like I know the pattern (or at least I think I do!) But i dont exactly know how to word it.


2) Evaluate the following sums:
1
1+8
1+8+27
1+8+27+64

Add two more lines to the list, and make a conjecture about the sums you obtained.


So when evaluted I get:
1
9
36
100

And I figured out that when you subtract the sum from the last number (as in 100-64 or 9-8), you get the answer from above. (100-64=36, 9-8=1) But I cant find the pattern really. I know that the next line will have to equal 100 when the last number in the sequence/question thing is subtracted from the answer you get, but other than that I just cant see it...

 

[stapel]

1) Make a conjecture about the next 3 terms in the sequence 1,5,2,8,3,11,... Explain your reasoning.

Split this into two:

. . . . .1, 2, 3,...

. . . . .5, 8, 11,...

Can you see a pattern now?

2) Evaluate the following sums:
1, 1+8, 1+8+27, 1+8+27+64
Add two more lines to the list, and make a conjecture about the sums you obtained.

What is 1^3? What is 2^3? What is 3^3?

What is the sum of 1? of 1 + 2? of 1 + 2 + 3? What is the formula?

What is the square of 1? of 1 + 2? of 1 + 2 + 3? So what is the formula?

Eliz.

 

[soroban]

Hello, K.ourt!

2) Evaluate the following sums:
1
1+8
1+8+27
1+8+27+64 . . Evidently you don't see a pattern here

Add two more lines to the list, and make a conjecture about the sums you obtained.


So when evaluated I get:
1
9
36
100 . . . . . and these numbers don't ring a bell

Do you see what they are adding? . . . cubes!

. . . . . .1³ . . . . . . . = . . . . . . 1 . . . . . . = . . 1
. . . .1³ + 2³ . . . . . = . . . . .1 + 8 . . . . .= . . 9
. . 1³ + 2³ + 3³ . . .= . . . 1 + 8 + 27 . . = . .36
1³ + 2³ + 3⁴ + 4³ .= .1 + 8 + 27 + 64 .= .100


Then the new two lines must be:

. . 1³ + 2³ + 3³ + 4³ + 5³ . . .= . . 1 + 8 + 27 + 64 + 125 . . . . = .225
1³ + 2³ + 3³ + 4³ + 5³ + 6³ .= .1 + 8 + 27 + 64 + 125 + 216 .= .441


Did you notice that the sums are square numbers? .1², 3², 6², 10², 15², 21². . .

If you can find the pattern for: 1, 3, 6, 10, 15, 21, ...
. . you can write a general formula for the sequence.