Thread: Conjectures, reasoning

Jacques noticed that they added an even number of blockes in each step. They started with 2, then added 4, and then added 6. He also noticed that the total number of blocks was related to the number of adjacent towers.

Number of towers. total number of blocks
1 2=1*2
2 2+4=6 =2*3
3 2+4+6=12 =3*4


Jacques concluded that the sum of the first n even numbers is n(n+1); that is,

2+4+6+...+2n=n(n+1)

a.)Jacques mother observed that 2+4+5+...+2n is an example of an arithmetic series and that the sum of the terms of an arithmetic series is determined as follows:

t1+t2+t3+...tn = n(t1+tn)/2

Prove Jacques conjecure using this formula. Is the proof an example of inductive reasoning or deductive reasoning? Explain.




I'm sorry it's long, but I don't know how to go about this question or anything, Details appreciated

Originally Posted by booper563

Jacques noticed that they added an even number of blockes in each step. They started with 2, then added 4, and then added 6. He also noticed that the total number of blocks was related to the number of adjacent towers.

Number of towers. total number of blocks
1 2=1*2
2 2+4=6 =2*3
3 2+4+6=12 =3*4


Jacques concluded that the sum of the first n even numbers is n(n+1); that is,

2+4+6+...+2n=n(n+1)

a.)Jacques mother observed that 2+4+5+...+2n is an example of an arithmetic series and that the sum of the terms of an arithmetic series is determined as follows:

t1+t2+t3+...tn = n(t1+tn)/2

Prove Jacques conjecure using this formula. Is the proof an example of inductive reasoning or deductive reasoning? Explain.

1. Show that Jacques activity gives rise to an arithmetic progression and identify the first and last terms after n steps. Now apply the AO sum formula.

Jaques reasoning is inductive as he is going from a finite set of observations to a general law.

Jacques mother's reasoning is deductive in that she is demonstrating deductivly that the AP law applies and then using the kniw result.

CB