Evaluate the following sums:
1
1+8
1+8+27
1+8+27+64
Add two more line to the list, and make a conjecture about the sums you obtained.
this is a pattern. if you look closely than you'll see each number is a product of another number cubed.
(1^3)+(2^3)+(3^3)... and so on. try this and see what you get.
thanks that's awesome. Do you just guess and try things to figure out conjectures?
What about this one?
Make a conjecture about the next 3 terms in the sequence 1,5,2,8,3,11....
Explain your reasoning.
no i just saw the pattern
okay educate me on what a conjecture is.
it is a mathematical statement that appears likely to be true, but has not been formerly proven true.
ahh. i see. okay so this one, is
1, 5, 2, 8, 3, 11
okay this one is a bit igreular, for the first number, 1, add 4, the second number is 5 and your going to subtract 3, the third number is 2 and your going to add six, the fourth number is 8 and your going to subtract 5. do you see the pattern?
1, (1+4)=
5, (5-3)=
2, (2+6)=
8, (8-5)=
3, (3+8)=
No I don't see the pattern. 4 3 6 5 8 is no pattern. but how do I write that as a conjecture?
well like i said it is kinda irregular. okay look first at the signs:
1, (1+4)=
5, (5-3)=
2, (2+6)=
8, (8-5)=
3, (3+8)=
it goes add, subtract, add, subtract and so on. the next step is to look real close at the numbers. we are starting with adding 4. then skip the next 5, we'll look at that later. so now we look at 2, and were adding six to that. how do you go to get from 4 to 6? basically every other number (starting with the first number in the sequence) you have to add the next greatest even number. in this case, starting with adding 4, skipping the 5, and adding 6 (the next greatest even number after 4) to the 2. now skip the next number in the sequence, the 8, and go to the 3. what do you do to that? add 8 becuase that is the next even number after 6. keep going like that
now look at the 5. to get 5 you had to add the 4 to the 1. to go the the 2 fromthe 5 your going to now do subtraction. remember what we established earlier, that one part of this pattern is adding, subtracting, adding, subtracting. so now we are on to the subtraction. here we are going to use the same method with the even numbers but this time it is going to be odd. how do you go from 5 to 2? you subtract 3. how now we skip the 2 becuase we know how to go from 2 to 8, (by adding 6 like we said earlier) and we try to figure out how to go from the 8 to the 3, which is subtracting 5 (which is the next greatest odd number after 3). i know this is confusing but hoppefully this is helping you a little. do you get it?
yes, thanks for the help. can you look at the other one i have up? maybe you can help me with understanding that. thanks again
of course. not a problem. and i would love to help you with however many you have. ask away.
of course. not a problem. and i would love to help you with however many you have. ask away.
of course. not a problem. and i would love to help you with however many you have. ask away.
I apologize for the repetition in my response. It seems there was a glitch.
Now, let's look at the sequence 1, 5, 2, 8, 3, 11. At first glance, it may appear irregular, but there could still be a pattern. When observing the numbers closely, we can see that the even-indexed terms (1st, 3rd, 5th) seem to follow a pattern of increasing by a certain amount, while the odd-indexed terms (2nd, 4th, 6th) seem to follow a different pattern of increasing or decreasing by another amount.
If we break down the sequence:
1, 5, 2, 8, 3, 11
We can see that the even-indexed terms increase by 3:
1 + 3 = 4
2 + 3 = 5
3 + 3 = 6
The odd-indexed terms, however, increase or decrease by a greater amount.
Starting from the 5, we subtract 3:
5 - 3 = 2
Then, from the 8, we subtract 4:
8 - 4 = 4
Finally, from the 11, we subtract 7:
11 - 7 = 4
So the odd-indexed terms actually decrease by consecutive odd numbers: 3, 4, 7.
Based on this pattern, we can make a conjecture about the next three terms in the sequence:
4 + 3 = 7
6 - 4 = 2
4 + 7 = 11
Therefore, we can conjecture that the next three terms in the sequence would be 7, 2, 11.
Please note that a conjecture is a statement that seems likely to be true based on observed patterns, but it is not proven to be true.