\(\(4+8+\dots+64=\sum_{n=2}^{6}{2^{n}}\)\)¶

|\(a_n=2^n\)|||||| |-|-|-|-|-|-|-|-|-| |\(a_n\):|2|4|8|16|32|64|128|256| |\(n\):|1|2|3|4|5|6|7|8|

Is this property True or False in general?¶

\(\(\sum_{n=1}^{4}{2*n}=2*\sum_{n=1}^{4}{n}=\underset{\text{These are equal by distributivity!}}{(2*1)+(2*2)+(2*3)+(2*4)=2(1+2+3+4)=\underset{\text{Don't simplify!}}{\cancel{2+4+6+8=20}}}\)\) These are equal by distributivity!

tldr¶

On the next 5 slides you will need to decide whether various properties of summation notation actually hold or are in fact false. To help with this, you should consider specific examples.

For instance, a sum can be generally written as: \(\sum_{n=1}^{k}{a_{n}}\)

For the specific sequence $a_{n}=2n+1 and \(k=5\), this sum would be \(\sum_{n=1}^{5}{2n+1}=3+5+7+9+11\)

1¶

Is the following a True property of summation notation? Explain in a sentence, using specific examples. \(\(\sum_{n=1}^{k}{c*a_{n}}=c*\sum_{n=1}^{k}{a_{n}}\)\)

2¶

Is the following a True property of summation notation? Explain in a sentence, using specific examples. \(\(\sum_{n=1}^{k}{a_n*b_n}=\left(\sum_{n=1}^{k}{a_{n}}\right)*\left(\sum_{n=1}^{k}{b_{n}}\right)\)\) \(\(\underset{(2*3)+(3*4)+(4*5)+(5*6)}{\sum_{n=1}^{4}{(n+1)*(n+2)}}\ne\underset{(2+3+4+5)*(3+4+5+6)}{\left(\sum_{n=1}^{4}{n+1}\right)*\left(\sum_{n=1}^{4}{n+2}\right)}\)\) False!

3¶

Is the following a True property of summation notation? Explain in a sentence, using specific examples. \(\(\sum_{n=1}^{k}{a_n+b_n}=\left(\sum_{n=1}^{k}{a_{n}}\right)+\left(\sum_{n=1}^{k}{b_{n}}\right)\)\) \(\(\underset{(2+3)+(3+4)+(4+5)+(5+6)}{\sum_{n=1}^{4}{(n+1)+(n+2)}}=\underset{(2+3+4+5)+(3+4+5+6)}{\left(\sum_{n=1}^{4}{n+1}\right)+\left(\sum_{n=1}^{4}{n+2}\right)}\)\)