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
Distribute the factor $\frac 34$, and then factor out $5^{k+1}.\;$ Your expression is equivalent to:
$$5^{k+1}\left(\frac 34 + 3\right) - \frac 34 = \frac {15}{4}5^{k+1} - \frac 34= \frac 34(5\cdot 5^{k+1} - 1) = \frac 34(5^{k+2} - 1)$$