Mathematical Investigations IV
Name
Mathematical Investigations IV
Iteration Forever
MORE SPECIAL SUMS
1.
Next Special Series: Sn 
n
 j2
j 1
Let's derive the formula for the sum of the first n perfect squares.
Complete the table below:
n
1
2
3
4
5
6
an = n2
Sn
Let us assume that the relationship (n, Sn) can be determined by a polynomial in n. Then,
we can use finite differences to determine the degree of the polynomial and then use our
calculators to derive the function.
a.
Using finite differences on the table above, what is the degree of the polynomial?
b.
Using your calculator or any other means you wish, derive the function:
Sn 
n
 j2 
j 1
100
2.
Find
a.
j
2
 j 2  2 j 
25
b.
j 1
j 1
Seq & Ser. 9.1
Rev. S07
Mathematical Investigations IV
Name
3.
Still One More Special Series: Sn 
n
 j3
j 1
Complete the table below:
n
1
2
3
4
5
6
an = n3
Sn
Let us assume once again that the relationship (n, Sn) can be determined by a polynomial
in terms of n.
a.
Using finite differences, what is the degree of the polynomial?
b.
Using your calculator or any other means you wish, derive the function:
Sn 
n
 j3 
j 1
More practice. Evaluate each of the following.
12
15
4.
 4k 3
5.
6.
 2k
k 1
3
j3
k 1
12
j
3
2
 5k  6

 2k 3  5k 2  6
25
7.
k 13
Seq & Ser. 9.2
Rev. S07
Mathematical Investigations IV
Name
Seq & Ser. 9.3
Rev. S07