OldTimer

Great job Alan ! Thanks...so helpful

Dear all thanks for all your help! What a great team you are! 

Wow...food for thought! Thanks.

Hi Alan thanks for your follow up......If I may bother you for a bit more clarification

>>The formulas i am following are as below:

Therefore, I have understood solutions obtained as far as angles go...

but

the logic of how you derived  extensions (as highlighted below in your answer) escapes me.

Thanks for your patience!

Hi Alan...great job thanks!

Prior to posting question and after long hours of problem solving I did obtain 36 deg and 108 degrees. My main issue was applying the standard formulas for the general angle...in this case cosine. Therefore I am now trying to see how I can reconcile those answers....hopefully I can solve !

All the best for Xmas of course and New Year....thanks again! 

Smooth as silk...Thanks a lot!

There must be some error in logic here:

 Hi Melody, Thanks for your update....you are very always helpful...I can vouch for that!

I have a partial solution which I will post later on tomorrow (loading is a bit tough)...but need to rework  as in desperation I worked back from the answer, but  still not perfect.      

Thanks Melody..that was a great help and an eye opener...regards! 

HI Cphill

I did not really understand exactly second part of question but I have interpreted it as follows:

Define Pattern :

S_{c = 1^{3 +} 3^{3 +} 5³..........+ (2n-1)³  =  n²(2n²-1)  (as per formula for sum of cubes for odd integers)}

Now Sum of cubes for n+(n+1)

S_{2n+1 = n²(2n²-1) + [ 2(n+1)-1]³}

= 2n⁴- n² + [2n+1]³

= 2n⁴- n²  + 8n³+ 12n² + 6n + 1

= (2n⁴ + 4n³ + 2n²) + (4n³ + 8n² + 4n) + (n² + 2n + 1)

= 2n²(n+1)^{2 +} 4n(n+1)² + (n+1)²

⁼ (n+1)²(2n² + 4n + 1)