# MATH 3600 - HW07

Date: ________________

In 1762 the arithmetician, Elie de Joncourt, published a small quarto volume of the first 19,999 numbers in the following sequence.

1, 3, 6, 10, 15, 21, 28, . . .

Exercise 1:  Copy and paste the first 7 numbers of this sequence into the Online Encyclopedia of Integer Sequences, and write the ID  of this sequence as well as the name and the immediate formulas in the block below.

Exercise 2:  In the next block generate the first 20 terms of the arithmetic sequence defined by $\alpha=5$ and $\beta=3$.  Use the recurrence relation.  Then in the subsequent block use the explicit formula to generate the first 20 terms of that arithmetic sequence.

alpha = 5 beta = 3 A1 = [alpha]
print 'A1',A1
A2 =
print 'A2',A2

Exercise 3:  Copy and paste the first 7 numbers of A2 into the Online Encyclopedia of Integer Sequences, and write the ID of this sequence in the block below.

An Arithmetic Series is a sequence of partial sums of an arithmetic sequence.  Given the arithmetic sequence [$a_0$, $a_1$, $a_2$, $a_3$, . . . , $a_n$] the corresponding arithmetic series is

[ $s_0 = a_0$,   $s_1 = a_0 + a_1$,   $s_2 = a_0 + a_1+a_2$,   $s_3 = \sum_{k=0}^3 a_k$,  . . . ,   $s_n = \sum_{k=0}^n a_k$ ]

Previously we saw that the recurrence relation for the Arithmetic Series is given by

$s_n = 2s_{n-1} - s_{n-2} + \beta$   with   $s_0 = \alpha$   and   $s_1 = \alpha+\beta$.

In addition the explicit formula for $s_n$ is given by

$s_n = (\alpha +[\alpha + n\beta])(n+1)/2$

Exercise 4:  In the next block use the partial sums definition to generate the first 20 terms of the Arithmetic Series based on the arithmetic sequence, A2.  In the subsequent block generate the Arithmetic Series using the two-term recurrence relation with the values of $\alpha$ and $\beta$ that defined A2.  In the third block generate the Arithmetic Series using the explicit formula.

S1 = [A2[0]]
S2 = [A2[0]]
S3 =
print 'S1:',S1 print 'S2:',S2 print 'S3:',S3

Exercise 5:  Copy and paste the first 7 numbers of S3 into the Online Encyclopedia of Integer Sequences, and write the ID of this sequence and the formula given next to the ID in the block below.

Exercise 6:  Is the formula given by OEIS the same as the explicit formula you used to compute S3.  If not, how is it different and what do you need to do to convert it to your formula?

Exercise 7:  Using Mathematical Induction, show that the explicit formula

$s_n = (\alpha +[\alpha + n\beta])(n+1)/2$

generates the terms of the sequence defined by the linear, constant coefficient, two-recurrence relation

$s_n = 2s_{n-1} - s_{n-2} + \beta$   with   $s_0 = \alpha$   and   $s_1 = \alpha+\beta$.

for all $0 \le n$.