__Directions__ : By trying the following examples, and more if necessary, (at least one more) try to make a conjecture (educated guess) inspired by the examples, as indicated. In each case, once you have a conjecture, say how confident you are that your conjecture is correct. (How much are you willing to bet on it?) Then, if you can, provide a completely convincing argument that your conjecture is correct or is incorrect, as appropriate.

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#2)

a. Examples: copy and complete: 1=1; 1+3=...;1+3+5=...;1+3+5+7=...;

b. Complete to make a conjecture: for n a natural number, 1+3+...+(2n+1)=...

c. Examples: copy and complete: 1=1; 1+8=...;1+8+27=...;1+8+27+64=...;

d. Complete to make a conjecture: for n a natural number, 1+8+...+n^3)=...

e. Say whether each of the following is or is not prime

- 1^2+1+41;
- 2^2+2+41;
- 3^2+3+41;
- 4^2+4+41;

f. make a conjecture: for n a natural number is n^2+n+41 is ...

g.express the even number as a sum of two prime numbers.

h. Complete the conjecture: Every even number ...

MY WORK:
a. 1=

1; 1+3=

4;1+3+5=

9;1+3+5+7=

16;

b. for n a natural number, 1+3+...+(2n+1)= ???

c. 1= 1; 1+8=

9;1+8+27=

36;1+8+27+64=

100...

d. for n a natural number, 1+8+...+n^3)= ???

e.

- 1^2+1+41= 43 is a prime #
- 2^2+2+41= 47 is a prime #
- 3^2+3+41= 53 is a prime #
- 4^2+4+41= 61 is a prime #

f. conjecture:

n^2+n+41 will result in a prime number.
g.

- 4= 2 + 2
- 6= 3 + 3
- 8= 5 + 3
- 10= 5 + 5

h. conjecture: Every even number can be the sum of two prime numbers.