Home > Problem of the Week > Archive List > Detail

<< Prev 1/27/2008 Next >>

How Clever Are You......

If A + B = 1 and A2 + B2 = 2, then what is A3 + B3?


If A + B + C = 1 and A2 + B2 + C2 = 2 and A3 + B3 + C3 = 3, then what is A4 + B4 + C4?

Want to make up the next one yourself...there is a pattern lurking here....

 


Hint: Don't rush to do substitutions and alot of algebraic manipulations....stop and think. It is possible to find the value of A3 + B3 without first finding the values of either A or B.

 


Solution Commentary: Specific to the first problem:

1 x 2 = (A+B)(A2 + B2) = A3 + B3 + AB(A+B)

But 1 = (A+B)(A+B) = A2 + B2 + 2AB = 2 + 2AB

This implies AB = - 1/2, which implies that A3 + B3 = 2 - (- 1/2) = 2 1/2

Feel free to go through all of the algebra to check this out...You should find that A = [1 SQRT(3)]/2, etc.

Now, you can try your hand at the second question.

Also, in the first question, is it helpful to note that A + B = 1 can be graphed as a line and A2 + B2 = 2 can be graphed as a circle....that intersect twice?