Math Test, Please Help

[b etor]

do your own work you lazy little punk! what happens when you get into the exams and cant do it hey?!

that's why he wants to see the work.

 

[Kozmos]

If you people are in love with math problems I'll might post some of my tests. They are damn driving me crazy.

 

[silver 2039]

yeh and u can do sum of mine.

 

[Atticus]

Sum?

At least I'm glad to help, but rather than doing the exercises it would be more helpful if people would give hints to solve them. I'm sure you've seen enough of examples allready. And usually it's a good idea to tell where the difficulty is.

 

[silver 2039]

1. 1+2+3=…+n= n(n+1)
2
a1= 1*2
a2=2*3
a3=3*4
a4=4*5
a5=5*6
a6=6*7
a7= 7*8

2.

Sn=a1+a2+a3+….+an

(a)

S1= a1
=1*2
=2

S2= a1+a2
=1*2+2*3
=8

S3= a1+a2+a3
=1*2+2*3+3*4
=20

S4= a1+a2+a3+a4
=1*2+2*3+3*4+4*5
=40

S5=a1+a2+a3+a4+a5
=1*2+2*3+3*4+4*5+5*6
=70

S6= a1+a2+a3+a4+a5+a6
=1*2+2*3+3*4+4*5+5*6+6*7
=112


(b) Sn= n(n+1)

c.

To find the conjecture

1 2
2 8
3 20
4 40
5 70
6 112




1/3x3+xx+2/3x

1/3n3+n2+2/3n

n (

Sn= 2+8+20+40+70+112 +n(n+1) =n(n+1)(n+2)
3
Let n=1

RHS=2 LHS= 1(1+1)(1+2)
3
=6
3
=2

Thus verified LHS=RHS

Let n=k

2+8+20+40+70+112+k(k+1)= k(k+1)(k+2)
3

Let n=k+1

2+8+20+40+70+112+k(k+1) +(k+1)(k+1+1)= (k+1)(k+1+1)(k+2+1)
3
(k+1) (k+2)(k+3) + (k+1)(k+2)= (k+1)(k+1+1)(k+2+1
3 3

(k+1)(k+2)(k+3) +3(k+1)(k+2)= (k+1)(k+1+1)(k+2+1
3 3

(k+1)(k+2) (k+3)
3

(d) 12+22+32+42+….+n(n+1)(n+2)=

Here try this. Because I'm hopelessly stuck. And this is due Friday. And worth 25% of my grade or so.

 

[Atticus]

Oh you really did mean sums I thought you just wrote 'sum' instead of 'some'.

It's a bit hard to read, but at first glance: are you proving the right thing? I suppose that you are asked to prove

1+2+3+....+n=n(n+1)/2,

and instead start to sum the right hand side, am I right?