U1=−1,U2=0,Un+2=6Un+1−9Un,ProveUn=(n−2)3n−1 So I done the first bit
Step 1 Prove true for n=1 and n = 2 U1=(1−2)30=1[br]U2=(2−2)31=0 Both true
Step 2 Assume true for n=k and n=k+1, if true for n=k and n=k+1 then true for n=k+2
Uk=(k−2)3k−1[br]Uk+1=(k−1)3k[br]ProveUk+2=(k)3k+1[br]Uk+2=6(k−1)3k−9(k−2)3k−1[br] ..... .... .... Step 1 + Step 2 proves true for all n by induction
I don't know what to do now, may someone help me please
You have to show that Uk+2=k(3k+1). You are correct so far and you can get to the required result from the step you're at.
I expanded it but i remember my teacher saying don't expand it unless you absolutely have to, and i can't tell if i have to or not like if there is another option
So would it be Uk+2=6(k−1)3k−9(k−2)3k−1[br]therefore[br]=[br]6(k−1)3k−32(k−2)3k−1[br][br]therefore[br]=3[2(k−1)k−3(k−2)3k−2][br][br]
Rather than taking factorising 3 in the last step, do you notice that you'll get in the first term 3×3k and in the second term there is a 32×3k remembering that ab×ac=ab+c
Also you've written it a bit wrong. It is uk+2=6(k−1)×3k−9(k−2)×3k−1
remember it's 3^k, not 3k, so you can't factorise 3 like that.
Rather than taking factorising 3 in the last step, do you notice that you'll get in the first term 3×3k and in the second term there is a 32×3k remembering that ab×ac=ab+c
Also you've written it a bit wrong. It is uk+2=6(k−1)×3k−9(k−2)×3k−1
remember it's 3^k, not 3k, so you can't factorise 3 like that.