rajshah428
New member
- Joined
- Oct 29, 2009
- Messages
- 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
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