Interesting Pattern?

Algebra Level 2

If a + b = 1 a + b = 1 ,

and a 2 + b 2 = 2 a^2 + b^2 = 2 ,

then what is a 3 + b 3 a^3 + b^3 ?

4 1.5 3 2 2.5 1

Winston Choo by Winston Choo , 46, Singapore

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1 solution

Jordan Cahn
Oct 16, 2018

First, ( a + b ) 2 = a 2 + 2 a b + b 2 1 = 2 + 2 a b − 1 2 = a b \begin{aligned} (a+b)^2 &= a^2 +2ab + b^2 \\ 1 &= 2 + 2ab \\ -\frac{1}{2} &= ab \end{aligned}

Using this fact, ( a + b ) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3 1 = a 3 + b 3 + 3 a b ( a + b ) 1 = a 3 + b 3 − 3 2 5 2 = a 3 + b 3 \begin{aligned} (a+b)^3 &= a^3 + 3a^2b + 3ab^2 + b^3 \\ 1 &= a^3 + b^3 + 3ab(a+b) \\ 1 &= a^3 + b^3 -\frac{3}{2} \\ \frac{5}{2} &= a^3 + b^3 \end{aligned}

So a 3 + b 3 = 5 2 = 2.5 a^3+b^3 = \frac{5}{2} = \boxed{2.5} .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

You can use a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 ) a^3+b^3=(a+b)(a^2-ab+b^2)

X X - 2 years, 7 months ago

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That would work too!

Jordan Cahn - 2 years, 7 months ago

And we can generate higher power sums using

a 2 n + b 2 n = ( a n + b n ) 2 − 2 × ( − 1 2 ) n a^{2n} + b^{2n} = (a^{n} + b^{n})^{2} - 2 \times \left(-\dfrac{1}{2}\right)^{n} and

a 2 n + 1 + b 2 n + 1 = ( a n + b n ) ( a n + 1 + b n + 1 ) − ( − 1 2 ) n a^{2n + 1} + b^{2n + 1} = (a^{n} + b^{n})(a^{n + 1} + b^{n + 1}) - \left(-\dfrac{1}{2}\right)^{n} .

Brian Charlesworth - 2 years, 7 months ago

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