How to Simplify a C1 Level Expression Involving Constants and Variables?

[CathyLou]

Show that http://www.artofproblemsolving.com/Forum/latexrender/pictures/4cb05ba8a17dcb9e2dcfb9ef1a98966a.gif where k is a constant to be found.

I know that a = 2 and that d = 1/2 and substituting these figires into Sn = (2a + (n-1) d) gives n/2 (4 + (n-1)/2) but I just can't see how to get from n/2 (4 + (n-1)/2) to the answer of 1/4n(n+7). I'd really appreciate it if someone could please explain.

Thank you.

Cathy

 

[AlephZero]

Try using

4 + (n-1)/2 = (8+n-1)/2 = (n+7)/2

 

[CathyLou]

Try using

4 + (n-1)/2 = (8+n-1)/2 = (n+7)/2

Thanks for replying but I still don't understand how you get from 4 + (n-1)/2 to (8+n-1)/2.

 

[cristo]

Thanks for replying but I still don't understand how you get from 4 + (n-1)/2 to (8+n-1)/2.

In order to put 4 over the denominator 2, you must first multiply it by 2. Let's consider a general example: $$\frac{a}{b}+\frac{c}{d}=\frac{ad+cb}{bd}$$

Now applying this to your case we have, strictly speaking, $$\frac{4}{1}+\frac{n-1}{2}=\frac{2(4)+1(n-1)}{2}=\frac{8+n-1}{2}$$

 

[CathyLou]

In order to put 4 over the denominator 2, you must first multiply it by 2. Let's consider a general example: $$\frac{a}{b}+\frac{c}{d}=\frac{ad+cb}{bd}$$

Now applying this to your case we have, strictly speaking, $$\frac{4}{1}+\frac{n-1}{2}=\frac{2(4)+1(n-1)}{2}=\frac{8+n-1}{2}$$

Thanks for your help.

I understand it now.

Cathy