Need Help Solving Polynomial Equation

Question

I'm working on an induction problem that basically boils down to this equation:

$$2(-1)^k+ 6(2^k)\left(-\frac{1}{2}\right)^{k+1} + (-1)^{k}=0$$

I'm fairly confident that the equation above is the solution to the problem, but I am unable to simplify it further in order to prove the case.

Any help breaking it down would be appreciated.

if you simplify $2^k$ and $1/2^{k+1}$ you will be almost there. — Maesumi, Oct 21 '13 at 20:29
This, http://i.imgur.com/6cjzQI6.png is what I'm trying to prove, using induction (P(k) -> P(k+1)) — fakeplasticandroid, Oct 22 '13 at 22:46

score 1 · Accepted Answer · answered Oct 21 '13 at 20:31

1

$$ 2(-1)^k + 6(2)^k (- \frac{1}{2})^{k+1} - (-1)^{k+1} =0 $$ Notice $(- \frac{1}{2})^{k+1} = (-1)^{k+1}(2)^{-k-1}$and multiply by $(-1)^{k+1}$ $$ -2 + 6(2)^k(2)^{-k-1}-1=0 $$ Simplify $$-3 + 6(2)^{-1} =0$$ so finally $$-3 + 3 =0$$ which is an identity, so your original equation is true for all values of $k$.

answered Oct 21 '13 at 20:31

TBrendle

3,253

Brilliant, thanks. I had tried to simplify (-1/2)^(k+1) to (-2)^(-k-1), but couldn't take it farther from there. – fakeplasticandroid Oct 22 '13 at 22:43

Need Help Solving Polynomial Equation

1 Answers1