Here’s a simple little maths problem for you to look at, if you like that sort of thing.
For what value of x does the following equation hold true?
x^n + 2.x^n+1 = x^n+3
Where x^3 is x cubed.
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x=-1. I did enjoy this little puzzle but I had to get out a pencil and paper probably because I have become so very old! If this is an incorrect answer I am even older than I first thought. Nice image of the Andromeda Galaxy group.
Blimey!! Well done – I wasn’t expecting that – and it wasn’t the answer I had 🙂 Can you find another solution for x? It is MUCH more difficult to find (but maybe not for you).
x=1+or – square root of 2 ? so 3 solutions to x in all.
You’re coming out with things I was not expecting here 🙂 The number I used to get this equality was in fact Phi, the Golden Ratio.
Hi Greg was concentrating on tea rather than the problem. Made a very silly mistake re the square root of 2 solutions please discard. I told you I was old. Have however realised a nice way of coming up with -1 and Phi. Reduce the general equation by taking n=0.
x^3-2x-1=0
Factorise
(x+1)(x^2-x-1)=0
Therefore either:
x+1 =0 or x=-1
OR
x^2-x-1=0 which is the formula for the golden ratio if you substitute Phi for x or is solvable using the quadratic formula giving two solutions (1+or-square root of 5)/2 presumably when minus it is the inverse of Phi. Bearing in mind I used to be an architect funny I should miss the Golden Ratio as a solution.
Thanks for keeping me entertained on a cold afternoon.
Regards George
Hi George – Well you seem to have done a proper job on that one, I’m very impressed.My maths is nowhere near as good as yours, I should have seen the x = -1 solution from the beginning. I too am old. Unfortunately this April, if I make it that far, I will hit 70. As someone who didn’t expect to get past 30 you can imagine the shock.
Regards Greg
Thank you Greg but I strongly doubt that my mathematical skills exceed yours but on the arts side I’m a real dab hand with the Crayola wax crayons It did occur to me however that in accordance with Mr Euler as e^i.pi=-1 then e^ipi is a solution of x for all n and that very simple equation then links the golden ratio to e, i and pi by being part of a defined set. It then occurred to me that there might be limits upon n (do these 3 ,(-1, phi ,1/phi,) values for x work for minus values of n etc). Maths is troubling because the more you think about a problem the more it gets away from you. Much like looking out into the Cosmos!
Best regards George
Yes – it is fatal to keep looking deeper into maths problems. I was looking at the Tower of Power I^I^I^I (tetration) and soon found myself in the World of fractals. Quite disturbing 🙂
Had to look up the ‘Tower of Power’ I think this is definitely above my ‘math pay grade’ . The first time I heard this expression, Frank Zappa used it to describe a joint.
Don’t worry about the ‘seventy barrier’ you will steam through if you can negotiate the complex plane for fun. I am in my seventy fifth year on this planet and I occasionally have to take a moment to remember why going up ladders is unwise. I have noticed that increasing age has slowed my thinking, rendered multitasking dangerous and brought the delights of ’ buffering’ to my on board memory. Otherwise all is well and there is still lots to find out!
George
You give me (a little) hope for the future, but this past year (69) was the first year I felt over 60. Before 2023 I felt like I was around 50.