\($4+8+\cdots +64=\sum _{n=2}^6{2}^n$\)¶

|$a_n=2^n$|||||| |-|-|-|-|-|-|-|-|-| |$a_n$:|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?¶

\($\sum _{n=1}^{4}2\ast n=2\ast \sum _{n=1}^{4}n=\underset{\text{These are equal by distributivity!}}{(2\ast 1)+(2\ast 2)+(2\ast 3)+(2\ast 4)=2(1+2+3+4)=\underset{\text{Don't simplify!}}{\cancel{2+4+6+8=20}}}$\) 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. \($\sum _{n=1}^{k}c\ast a_n=c\ast \sum _{n=1}^k{a}_n$\)

2¶

Is the following a True property of summation notation? Explain in a sentence, using specific examples. \($\sum _{n=1}^k{a}_n\ast b_n=\left(\sum _{n=1}^k{a}_n\right)\ast \left(\sum _{n=1}^k{b}_n\right)$\) \($\underset{(2\ast 3)+(3\ast 4)+(4\ast 5)+(5\ast 6)}{\sum _{n=1}^4(n+1)\ast (n+2)}\ne \underset{(2+3+4+5)\ast (3+4+5+6)}{\left(\sum _{n=1}^{4}n+1\right)\ast \left(\sum _{n=1}^{4}n+2\right)}$\) False!

3¶

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