quick question

**zpwnchen** · November 20th 2009, 06:05 AM

how could this one $(-3)(2)^{n-1} + 2^n(-1)$ simplify to $-2^{n+2}+3$ ?

**aman_cc** · November 20th 2009, 06:16 AM

Originally Posted by zpwnchen

how could this one $(-3)(2)^{n-1} + 2^n(-1)$ simplify to $-2^{n+2}+3$ ?

It doesn't. I'm getting
$-2^{n+2}+3.2^{n-1}$

**zpwnchen** · November 20th 2009, 06:18 AM

Thanks! but i do not know why it was deduced to $-2^{n+2}+3$ in solution manual.

**Plato** · November 20th 2009, 06:26 AM

Originally Posted by zpwnchen

how could this one $(-3)(2)^{n-1} + 2^n(-1)$ simplify to $-2^{n+2}+3$ ?

It is not true for $n=2$ . Is it?

**zpwnchen** · November 20th 2009, 07:31 AM

it was consistently true for all n terms.

$a_n = 2a_{n-1} -3 , a_0=-1$

$(-3)(2)^{n-1} + 2^n(-1)$ --> $-2^{n+2}+3$

**Defunkt** · November 20th 2009, 08:28 AM

$\forall n \geq 2, \ -3 \cdot (2)^{n-1} + -2^n \equiv 0 (mod 2)$
But $-2^{n+2}+3 \equiv 1 (mod 2)$

So it is obviously not true.

**zpwnchen** · November 20th 2009, 09:14 AM

Yeh! that's right!

I was just keep checking it " $-2^{n+2}+3$ " against " $a_n = 2a_{n-1} -3 , a_0=-1$ " for n terms and it was true.

So " $(-3)(2)^{n-1} + 2^n(-1)$ " is not right! I suppose it must be something wrong in somewhere.

Thank you!

**Plato** · November 20th 2009, 09:32 AM

Originally Posted by zpwnchen

Yeh! that's right!

I was just keep checking it " $-2^{n+2}+3$ " against " $a_n = 2a_{n-1} -3 , a_0=-1$ " for n terms and it was true.

So " $(-3)(2)^{n-1} + 2^n(-1)$ " is not right! I suppose it must be something wrong in somewhere.

For the sequence $a_n = 2a_{n-1} -3 , a_0=-1$ the function $f(n)=-2^{n+2}+3$ does give the terms of the sequence for $n=1,2,\cdots$ .

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