Mathematical Induction?

Question

How many ways can six pins be arranged around a circular hat band?

Use mathematical induction to prove that 1^2+2^2+3^2+...+n^2=n(n+1)(2n+1_/6 for all positive integral values of n.

Express 23+29+35+42+47 using sigma notation.

Answer 1

PROBLEM 1:

If 6 pins were in a line, there would be 720 permutations, but because you are putting them around a hat band, you need to divide this by 6.

The answer is 120 ways.

PROBLEM 2:

Prove that:
1² + 2² + ... + n² = n(n+1)(2n+1)/6 for all positive integer values of n.

Part 1: Prove the base case for n = 1:
1² = 1(2)(2(1) + 1) / 6
1² = 1(2)(3) / 6
1 = 6/6
1 = 1

Part 2: Assume it is true for n, prove it for n+1.
1² + 2² + ... + n² + (n+1)²

Using the case for 1² + ... + n²:
= n(n+1)(2n+1)/6 + (n+1)²
= n(n+1)(2n+1)/6 + 6(n+1)²/6
= [ n(n+1)(2n+1) + 6(n+1)² ] / 6

Factor out (n+1):
= (n + 1)[ n(2n + 1) + 6(n+1) ] / 6

Expand items in brackets:
= (n + 1)[ 2n² + n + 6n + 6 ] / 6
= (n + 1)[ 2n² + 7n + 6 ] / 6

Factor:
= (n + 1)(n + 2)(2n + 3) / 6

Rewrite in terms of n+1:
= (n+1)((n+1) + 1)(2(n+1) + 1) / 6

This is the correct format for n+1.

Therefore by induction, this is true for all positive integers values of n.

PROBLEM 3:

Are you sure you don't mean:
23+29+35+*41*+47

It looks like you add 6 to each term.

a(1) = 17 + 6 = 23
a(2) = 17 + 2*6 = 29
a(3) = 17 + 3*6 = 35
a(4) = 17 + 4*6 = 41 <-- *
a(5) = 17 + 5*6 = 57

S(n) = sum n = 1 to 5 [ 17 + 6n ]

Answer 2

if you were only to consider a straight line then you would have 6! = 720 permutations for the six pins. however, since they are in a circle then each of these permutations could be repeated 6 times about the circle so you have 6!/6 = 5! = 120 permutations

----

this sum can be written as:

n
∑ i² = n * (n + 1) *(2n + 1) /6
i = 1

show true for the base case n = 1

1² = 1 * 2 * 3 / 6 = 6/6 = 1

assume this is true for n = k

show it is true for n = k+1

k+1
∑ i² =
i = 1

(k + 1)² + k * (k + 1) * (2k + 1)/6

(6k² + 12k + 6 + 2k³ + k² + 2k² + k) / 6

(2k³ + 9k² + 13k + 6) / 6

(k + 1) * ((k + 1) + 1) * (2 (k + 1) + 1) / 6

thus proved by induction.


-- --- --- ---

23+29+35+42+47 = 176

4
{∑ (23 + 6i) } + 1
i = 0

= 176

Answer 3

Assuming y ou meant 41 instead of 42
∑{i=1 to 5} (17 + 6*i)

Answer 4

six pins can be arranged in 720 ways
6 choices for the first place
5 choices for the second place
4 choices for the third place, etc
so 6*5*4*3*2*1= 720


can't help you on the other 2