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Induction Problem (Polya)

  • #1
93
9

Homework Statement



Consider the table:

1 = 0 + 1
2 + 3 + 4 = 1 + 8
5 + 6 + 7 + 8 + 9 = 8 + 27
10 + 11 + 12 + 13 + 14 + 15 + 16 = 27 + 64

Guess the general law suggested by these examples, express it in suitable mathematical notation, and prove it.

Homework Equations


[/B]
It's clear that if [itex]n[/itex] is "row" number, then the right-hand side is [itex]n^3 + (n+1)^3[/itex]. The left-hand side is in arithmetic procession, and it can be expressed as [tex][(n+1)^2 + 1] + [(n+1)^2 + 2] + ... + (n+1)^2[/tex]

The Attempt at a Solution


[/B]
From the equations above, my guess at the general solution is [tex][(n+1)^2 + 1] + [(n+1)^2 + 2] + ... + (n+1)^2=n^3 + (n+1)^3[/tex] I verified that this is correct from the back of the book, but I'm having trouble proving it. I've been trying to use induction. The base case holds. Then we want to show that
[tex][((n+1)+1)^2 + 1] + [((n+1)+1)^2 + 2] + ... + (n+1)^2=(n+1)^3 + ((n+1)+1)^3[/tex] I'm not sure where to go from here. I tried a few algebraic manipulations, but they didn't seem promising.

SOURCE: Mathematics and Plausible Reasoning by George Polya. Chapter 1, question 2.
 
Last edited:

Answers and Replies

  • #2
Math_QED
Science Advisor
Homework Helper
2019 Award
1,400
518

Homework Statement



Consider the table:

1 = 0 + 1
2 + 3 + 4 = 1 + 8
5 + 6 + 7 + 8 + 9 = 8 + 27
10 + 11 + 12 + 13 + 14 + 15 + 16 = 27 + 64

Guess the general law suggested by these examples, express it in suitable mathematical notation, and prove it.

Homework Equations


[/B]
It's clear that if [itex]n[/itex] is "row" number, then the right-hand side is [itex]n^3 + (n+1)^3[/itex]. The left-hand side is in arithmetic procession, and it can be expressed as [tex][(n+1)^2 + 1] + [(n+1)^2 + 2] + ... + (n+1)^2[/tex]

The Attempt at a Solution


[/B]
From the equations above, my guess at the general solution is [tex][(n+1)^2 + 1] + [(n+1)^2 + 2] + ... + (n+1)^2=n^3 + (n+1)^3[/tex] I verified that this is correct from the back of the book, but I'm having trouble proving it. I've been trying to use induction. The base case holds. Then we want to show that
[tex][((n+1)+1)^2 + 1] + [((n+1)+1)^2 + 2] + ... + (n+1)^2=(n+1)^3 + ((n+1)+1)^3[/tex] I'm not sure where to go from here. I tried a few algebraic manipulations, but they didn't seem promising.
Indeed, induction is often used to proof things like this. However, you might want to use the following formula: https://en.wikipedia.org/wiki/Arithmetic_progression#Sum
 
  • #3
93
9

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