Determining the value of summations using combinatorial mathematics

This is from my book on discrete and combinatorial mathematics, in the chapter about combinations (just for clarity, this stuff Combination - Wikipedia, the free encyclopedia , but you probably new that). Considering the chapter it is from, simply filling it in shouldn't be the way they expect you to solve it. I probably have to use some sort of binominal coefficients. No idea how to apply it in this case, though...

It shouldn't be too hard, considering it is from the beginning of the book (and from a first year's course), but I simply have no idea.

All the exercise says is this: "Determine the value of each the following summations"

a)

6

Σ (i^2+1)

i=1

b)

2

Σ (j^3-1)

j=-2

c)

10

Σ (1+(-1)^i)

i=0

Re: Determining the value of summations using combinatorial mathematics

Quote:

Originally Posted by

**Amanoo** This is from my book on discrete and combinatorial mathematics, in the chapter about combinations

All the exercise says is this: "Determine the value of each the following summations"

a)

6

Σ (i^2+1)

i=1

b)

2

Σ (j^3-1)

j=-2

c)

10

Σ (1+(-1)^i)

i=0

These are such well known sums, why use anything else?

$\displaystyle \sum\limits_{k = 1}^n {k^2 } = \frac{{n(n + 1)(2n + 1)}}{6}\;\& \;\sum\limits_{k = 1}^n {k^3 } = \left[ {\frac{{n(n + 1)}}{2}} \right]^2 $

Just break up those sums.

Re: Determining the value of summations using combinatorial mathematics

Quote:

Originally Posted by

**Amanoo** This is from my book on discrete and combinatorial mathematics, in the chapter about combinations (just for clarity, this stuff

Combination - Wikipedia, the free encyclopedia , but you probably new that). Considering the chapter it is from, simply filling it in shouldn't be the way they expect you to solve it. I probably have to use some sort of binominal coefficients. No idea how to apply it in this case, though...

It shouldn't be too hard, considering it is from the beginning of the book (and from a first year's course), but I simply have no idea.

All the exercise says is this: "Determine the value of each the following summations"

a)

6

Σ (i^2+1)

i=1

b)

2

Σ (j^3-1)

j=-2

c)

10

Σ (1+(-1)^i)

i=0

The summation symbol is just a for loop (if that helps)

Here is the first one

It is just telling us to plug in i=1, i=2, i=3, i=4 and so forth until we get to i=6

$\displaystyle \sum_{i=1}^{6}(i^2+1)=(1^2+1)+(2^2+1+(3^2+1)+(4^2+ 1)+(5^2+1)+(6^2+1)$

Now just simplfy and add up all of the numbers.

Re: Determining the value of summations using combinatorial mathematics

Do you really think I should merely enter the numbers and add them up? It just seems too easy, especially since the chapter they're from isn't about summations at all. It is about combinations.

Re: Determining the value of summations using combinatorial mathematics

Quote:

Originally Posted by

**Amanoo** Do you really think I should merely enter the numbers and add them up? It just seems too easy, especially since the chapter they're from isn't about summations at all. It is about combinations.

I don't. But neither do I think combinatorics are necessary.

$\displaystyle \sum\limits_{k = 1}^6 {\left( {k^2 + 1} \right)} = \sum\limits_{k = 1}^6 {\left( {k^2 } \right)} + \sum\limits_{k = 1}^6 {\left( 1 \right)} = \frac{{6(6 + 1)(2 \cdot 6 + 1)}}{2} + 6$