[SOLVED] Proof by induction help needed (urgently)

**suf** · June 9th 2010, 03:19 PM

This is question 4 on June 2008 Further Pure 1 paper OCR (downloadable for OCR website) OCR > Qualifications > By type > AS/A Level GCE (current) > Mathematics > Mathematics > All documents

The question goes:
The Matrix A is given by A =

3, 1,
0, 1,

prove by induction that for n>1

A^n =

3^n, 0.5(3^n -1),
0, 1,

I just dont get how 3^k + 0.5(3^k -1) becomes 0.5(3^k+1 -1)

Thanks

**slider142** · June 9th 2010, 04:15 PM

Originally Posted by suf

...
I just dont get how 3^k + 0.5(3^k -1) becomes 0.5(3^k+1 -1)

Thanks

Simplify the expression: $3^k + \frac{1}{2} 3^k - \frac{1}{2}$

**suf** · June 10th 2010, 01:43 AM

Thats what im having probelms with

**Prove It** · June 10th 2010, 02:11 AM

Originally Posted by slider142

Simplify the expression: $3^k + \frac{1}{2} 3^k - \frac{1}{2}$

$3^k + \frac{1}{2}\cdot 3^k - \frac{1}{2} = \left(1 + \frac{1}{2}\right)3^k - \frac{1}{2}$

$= \frac{3}{2}\cdot 3^k - \frac{1}{2}$

$= \frac{1}{2}\cdot 3^{k + 1} - \frac{1}{2}$

$= \frac{1}{2}(3^k - 1)$ .

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