Series and Induction help needed alot

rajshah428

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Oct 29, 2009
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18
Description:
The Objective of this task is to investigate patterns and formulate conjectures (an educated guess) about numerical series. It is expected that students can recall that 1+2+3+...+n = n(n+1) / 2.

Method:
1. Consider the sequence {a*n}^infinity where a*1 = 1 . 2
a*2 = 2 . 3
a*3 = 3 . 4
a*4 = 4 . 5
.
.

Find an expression for a*n, the general term in the sequence.


2. Consider the series S*n = a*1 + a*2 + a*3 + ... + a*n where a*k is defined as above.

a. Determine several values of S*k, including including S*1,S*2,S*3,S*4,...S*6 and note observations.

b. Formulate a conjecture for a general expression for S*n

c. Using the above result, calculate 1^2 + 2^2 + 3^2 + 4^2 + ... + n^2


3. Consider T*n = 1 . 2 . 3 + 2 . 3 . 4 + 3 . 4 . 5 + ... + n(n+1)(n+2)

a. Determine several values of T*k, including T*1, T*2, T*3, T*4, ... T*6 and note observations.

b. Formulate a conjecture for a general expression for T*n

c. Using the above result, calculate 1^3 + 2^3 + 3^3 + 4^3 + .... n^3


4. Consider U*n = 1 . 2 . 3 . 4 + 2 . 3 . 4 . 5 + 3 . 4 . 5 . 6 + ... + n(n+1)(n+2)(n+3)

a. Determine several values of U*k, including U*1, U*2, U*3, U*4, ... ,U*6 and note observations.

b. Formulate a conjecture for a general expression for U*n

c. Using the above result, calculate 1^4 + 2^4 + 3^4 + 4^4 ... + n^4


5. With the patterns noted above, can you formulate a conjecture for the series

1^K + 2^k + 3^k + 4^k + ... + n^k ?
--------------------------------------…
Note '^' = power of
e.g: T*1 means the 1 is at the bottom right of T (below but part of number)
e.g: 1 . 2 = 1*2 (1 multiplied by 2)

I dont want all the answers, just working out for a few and how to get started, can i use graphs if yes, where. Thnx
 
rajshah428 said:
Description:
The Objective of this task is to investigate patterns and formulate conjectures (an educated guess) about numerical series. It is expected that students can recall that 1+2+3+...+n = n(n+1) / 2.

Method:
1. Consider the sequence {a*n}^infinity where a*1 = 1 . 2
a*2 = 2 . 3
a*3 = 3 . 4
a*4 = 4 . 5
.
.

Find an expression for a*n, the general term in the sequence.


2. Consider the series S*n = a*1 + a*2 + a*3 + ... + a*n where a*k is defined as above.

a. Determine several values of S*k, including including S*1,S*2,S*3,S*4,...S*6 and note observations.

b. Formulate a conjecture for a general expression for S*n

c. Using the above result, calculate 1^2 + 2^2 + 3^2 + 4^2 + ... + n^2


3. Consider T*n = 1 . 2 . 3 + 2 . 3 . 4 + 3 . 4 . 5 + ... + n(n+1)(n+2)

a. Determine several values of T*k, including T*1, T*2, T*3, T*4, ... T*6 and note observations.

b. Formulate a conjecture for a general expression for T*n

c. Using the above result, calculate 1^3 + 2^3 + 3^3 + 4^3 + .... n^3


4. Consider U*n = 1 . 2 . 3 . 4 + 2 . 3 . 4 . 5 + 3 . 4 . 5 . 6 + ... + n(n+1)(n+2)(n+3)

a. Determine several values of U*k, including U*1, U*2, U*3, U*4, ... ,U*6 and note observations.

b. Formulate a conjecture for a general expression for U*n

c. Using the above result, calculate 1^4 + 2^4 + 3^4 + 4^4 ... + n^4


5. With the patterns noted above, can you formulate a conjecture for the series

1^K + 2^k + 3^k + 4^k + ... + n^k ?
--------------------------------------…
Note '^' = power of
e.g: T*1 means the 1 is at the bottom right of T (below but part of number)
e.g: 1 . 2 = 1*2 (1 multiplied by 2)

I dont want all the answers, just working out for a few and how to get started, can i use graphs if yes, where. Thnx

First one is pretty straight forward - what did you get?

Please show us your work, indicating exactly where you are stuck - so that we know where to begin to help you.

DUPLICATE POST
http://answers.yahoo.com/question/index ... 708AAKyzfc
 
thnx...so for question 2 do they mean that a*k = n . (n+1) ???
and so S*k = i am not sure how you go about with 2a, any suggestions?
 
rajshah428 said:
thnx...so for question 2 do they mean that a*k = n . (n+1) ???
and so S*k = i am not sure how you go about with 2a, any suggestions?
S_1 = 1*2
S_2 = 2 + 2*3 = 8
S_3 = 8 + 3*4 = 20
S_4 = 20 + 4*5 = 40 .....and so on
 
So
S1 = 2
S2 = 8
S3 = 20
S4 = 40
S5 = 60

Bt for observations, i tried taking common differences but there is no pattern it goes like 6, 12, 20, 20. So i dont know if there are any observations. Since this a geometric series should i use any formulas? like the sum of nth term?

for 2b. A conjecture is an educated guess, but since i dont know the differences how can i make a general expression for T_N?
 
Srry for 2a i think i figured out sum stuff that went wrong. It shud be:
S1=2
S2=8
S3=20
S4=40
S5=70
And i found out there is a common difference, in the third time time u take the difference of those numbers which is 2.
Is there any special formula to do it because i did it in head mines has to mathematical and are there any other observations, except for the difference. Thnx alot.
Now, 2b......how to get a general expression, i know the third common difference is 2, does that help?
 
rajshah428 said:
Srry for 2a i think i figured out sum stuff that went wrong. It shud be:
S1=2
S2=8
S3=20
S4=40
S5=70
And i found out there is a common difference, in the third time time u take the difference of those numbers which is 2.
Is there any special formula to do it because i did it in head mines has to mathematical and are there any other observations, except for the difference. Thnx alot.
Now, 2b......how to get a general expression, i know the third common difference is 2, does that help ? <<< Yes ... Now find the fitting polynomial considering all the differences...
 
errmm...so let give it a try is it S_N = 2x^3 + 6x^2 + 6x???? i am sure its wrong...how shud i do it?
and the polynomial is the general expression?
 
Start with

S_N = Ax[sup:2gsy2fw2]3[/sup:2gsy2fw2] + Bx[sup:2gsy2fw2]2[/sup:2gsy2fw2] + Cx

then

S_1 = A + B + C = 2
S_2 = 8A + 4B + 2C = 8
S_3 = 27A + 9B + 3C = 20

Solve for A, B & C.
 
You need to find that (A, B & C) from those three equations (of S_1, S_2 & S_3).
 
okay so i got A = 1/3 B = 1 C = 2/3
so the general expression is 1/3x^3 + x^2 + 2/3x
so it can be written as x^3 + 3x^2 + 2x
so that is the conjecture.

now for 2c. how do you do it?
 
rajshah428 said:
c. Using the above result, calculate 1^2 + 2^2 + 3^2 + 4^2 + ... + n^2
That's simply the sum of squares: n(n+1)(2n+1) / 6
 
That's simply the sum of squares: n(n+1)(2n+1) / 6
okay..i am not sure if tht helps me..could you explain more..so to calculate 2c do i need to substitue the general expression 1/3x^3 + x^2 + 2/3x into n???pls explain how to this question
 
Using induction, show true for k=1:

\(\displaystyle \frac{1(1+1)(2(1)+1)}{6}=1\)......true.

Assume true for \(\displaystyle P_{k}\) and show true for \(\displaystyle P_{k+1}\)

\(\displaystyle 1^{2}+2^{2}+.....+k^{2}+(k+1)^{2}=\frac{k(k+1)(2k+1)}{6}+(k+1)^{2}\)

But, \(\displaystyle \frac{k(k+1)(2k+1)}{6}+(k+1)^{2}=\frac{(k+1)(k+2)(2(k+1)+1)}{6}\)

And it is proven.
 
I didnt quite understand the last part of P_k+1<<<< for that you are trying to prove. But the question says you have to calculate so how is that the answer (Q 2c.) ???
 
how did you get n(n+1)(2n+1) / 6, if we square the sides, shoudnt it be n^2+n^2+1+2n
okay..sorry now i understand galactus, the second part is simply proving it further. So when they say 'using the above result' they mean '1 + 2 + 3 + ... + n = n(n+1)/2

So for Question 3a.
T_1 = 6
T_2 = 6 + 24 = 30
T_3 = 30 + 60 = 90
T_4 = 90 + 120 = 210
T_5 = 210 + 210 = 420
T_6 = 420 + 336 = 756
^^^^^^^^^^^^^^^^^^^ are all of them correct, someone said they are wrong, btw i havent taken n(n+1)(n+2) into consideration because i did not know to use it, so can you tell me where i am going wrong and how to do it
 
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