Proof by induction, simplification step

Question

i have to prove (3/4)(5^(k+2) -1)

I have so far (after using inductive hypothesis etc): (3/4)(5^(K+1) -1) +3*5^(K+1)

I can't seen to find a useful common factor to simplify although i'm sure it would be 3*5^(k+1)

any help would be appreciated! Also, sorry about the styling i can't manage to find how to style the mathsiness and its been a while since i used LateX :O

What do you have to prove about $\left(\frac{3}{4}\right)\left(5^{k+2}-1 \right)$? — André Nicolas, Jun 03 '14 at 15:57
By the way, here is a mathjax tutorial if you're interested :) http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference — Leo, Jun 03 '14 at 15:59

amWhy · Accepted Answer · 2014-06-03T16:04:33.080

1

Distribute the factor $\frac 34$, and then factor out $5^{k+1}.\;$ Your expression is equivalent to:

$$5^{k+1}\left(\frac 34 + 3\right) - \frac 34 = \frac {15}{4}5^{k+1} - \frac 34= \frac 34(5\cdot 5^{k+1} - 1) = \frac 34(5^{k+2} - 1)$$

edited Jun 03 '14 at 16:04

answered Jun 03 '14 at 15:58

amWhy

209,954

yep! thanks i just solved thanks to your help! much appreciated, im a bit rusty with exponentials :p – pablo Jun 03 '14 at 16:04
You're welcome! – amWhy Jun 03 '14 at 16:05

Proof by induction, simplification step

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