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by djmathman, Jun 27, 2013, 8:04 PM
The following six exercises are from an American high school algebra textbook from 1975 after a section on summation notation.
The following six exercises are from an American high school Algebra II textbook from 2007 in various "Challenge and Extend" sections in a chapter on sequences and series.
Do you notice a difference?
Problem 10 wrote:
Show that ![$\displaystyle\sum_{j=1}^n a_jb_j=\tfrac14\left[\sum_{j=1}^n(a_j+b_j)^2-\sum_{j=1}^n(a_j-b_j)^2\right]$](//latex.artofproblemsolving.com/9/d/8/9d8840c562bf1c11e822e04ee076e8fc93203da2.png)
.
![$\displaystyle\sum_{j=1}^n a_jb_j=\tfrac14\left[\sum_{j=1}^n(a_j+b_j)^2-\sum_{j=1}^n(a_j-b_j)^2\right]$](http://latex.artofproblemsolving.com/9/d/8/9d8840c562bf1c11e822e04ee076e8fc93203da2.png)
.Problem 17 wrote:
Compute 
. [Hint: Consider 
where 
.]
. [Hint: Consider 
where 
.]Problem 20 wrote:
Use the relation ![\[\sum_{j=0}^nj^3=\sum_{j=0}^n(n-j)^3\]](//latex.artofproblemsolving.com/4/3/5/4357b5825c73567acecea6592ea6b3b64cc1262c.png)
to obtain the formula for 
in another way.
![\[\sum_{j=0}^nj^3=\sum_{j=0}^n(n-j)^3\]](http://latex.artofproblemsolving.com/4/3/5/4357b5825c73567acecea6592ea6b3b64cc1262c.png)
to obtain the formula for 
in another way.Problem 21 wrote:
Prove that no matter how large 
is, ![\[1+\dfrac1{2^2}+\dfrac1{3^2}+\cdots+\dfrac1{n^2}<2.\]](//latex.artofproblemsolving.com/a/5/a/a5a84adc6c3a7096e6be883c5d6d5935ae07ede5.png)
![$\left[\text{Hint: }\dfrac1{j^2}<\dfrac1{j(j-1)}\text{ for }j\geq 2.\right]$](//latex.artofproblemsolving.com/4/a/f/4af3fde7faa32ea3ed8adbc0f9dba54453c1b2e5.png)
is, ![\[1+\dfrac1{2^2}+\dfrac1{3^2}+\cdots+\dfrac1{n^2}<2.\]](http://latex.artofproblemsolving.com/a/5/a/a5a84adc6c3a7096e6be883c5d6d5935ae07ede5.png)
![$\left[\text{Hint: }\dfrac1{j^2}<\dfrac1{j(j-1)}\text{ for }j\geq 2.\right]$](http://latex.artofproblemsolving.com/4/a/f/4af3fde7faa32ea3ed8adbc0f9dba54453c1b2e5.png)
Problem 23 wrote:
Let 
and 
for 
. Prove
![\[\sum_{i=1}^nA_ib_i=A_nB_n-\sum_{i=1}^{n-1}a_{i+1}B_i.\]](//latex.artofproblemsolving.com/2/e/d/2edbebcb93aa7e67f6e2486eb434167aed8f729f.png)
and 
for 
. Prove![\[\sum_{i=1}^nA_ib_i=A_nB_n-\sum_{i=1}^{n-1}a_{i+1}B_i.\]](http://latex.artofproblemsolving.com/2/e/d/2edbebcb93aa7e67f6e2486eb434167aed8f729f.png)
Problem 24 wrote:
Find ![\[\dfrac3{(1\cdot 2)^2}+\dfrac5{(2\cdot 3)^2}+\dfrac7{(3\cdot 4)^2}+\cdots+\dfrac{2n+1}{[n(n+1)]^2}.\]](//latex.artofproblemsolving.com/5/b/f/5bf6f40f694e7f4c4e6b180fc880e0a65dcdeb68.png)
![\[\dfrac3{(1\cdot 2)^2}+\dfrac5{(2\cdot 3)^2}+\dfrac7{(3\cdot 4)^2}+\cdots+\dfrac{2n+1}{[n(n+1)]^2}.\]](http://latex.artofproblemsolving.com/5/b/f/5bf6f40f694e7f4c4e6b180fc880e0a65dcdeb68.png)
The following six exercises are from an American high school Algebra II textbook from 2007 in various "Challenge and Extend" sections in a chapter on sequences and series.
Lesson 12-1, Problem 56 wrote:
Write an explicit rule for the following sequence and find the 10th term.
![\[-2,6,-12,20,-30,\ldots\]](//latex.artofproblemsolving.com/1/7/7/17785914b14188c1d9acb7881902f3c2df2e4269.png)
![\[-2,6,-12,20,-30,\ldots\]](http://latex.artofproblemsolving.com/1/7/7/17785914b14188c1d9acb7881902f3c2df2e4269.png)
Lesson 12-2, Problem 60 wrote:
Prove the following summation property for the sequences 
and 
.
![\[\sum_{k=1}^n(a_k+b_k)=\sum_{k=1}^na_k+\sum_{k=1}^nb_k.\]](//latex.artofproblemsolving.com/8/d/2/8d2e08450eb732487b352b0171b3cfd82cdf524a.png)
and 
.![\[\sum_{k=1}^n(a_k+b_k)=\sum_{k=1}^na_k+\sum_{k=1}^nb_k.\]](http://latex.artofproblemsolving.com/8/d/2/8d2e08450eb732487b352b0171b3cfd82cdf524a.png)
Lesson 12-3, Problem 60 wrote:
The sum of three consecutive terms of an arithmetic sequence is 
. If the product of these terms is 
, what are the terms?
. If the product of these terms is 
, what are the terms?Lesson 12-3, Problem 61 wrote:
What does 
mean and for what arithmetic sequences is it true?
mean and for what arithmetic sequences is it true?Lesson 12-4, Problem 60 wrote:
The sum of three consecutive terms of a geometric sequence is 
. If the product of these terms is 
, what are the terms?
. If the product of these terms is 
, what are the terms?Lesson 12-5, Problem 68 wrote:
Write 
as a fraction in simplest form.
as a fraction in simplest form.Do you notice a difference?
This post has been edited 2 times. Last edited by djmathman, Sep 5, 2013, 1:29 AM
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