# Math Project on Series And Induction

#### rajshah428

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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 ?
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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

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