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Question on proof by induction for Inductive Sequences

Screenshot 2024-04-07 001415.pngScreenshot 2024-04-07 001458.png


Ok so on the 2nd part how on earth would they know that is the answer?? Is it just trial and error or am i missing something???
Reply 1
Original post by 43Explosion
Screenshot 2024-04-07 001415.pngScreenshot 2024-04-07 001458.png
Ok so on the 2nd part how on earth would they know that is the answer?? Is it just trial and error or am i missing something???

There is a hint in i) so
1/2(u_4 - 1) = 3^3
and u_2 and u_3 follow the same pattern and youve worked out their values. Also the term to term rule is roughly multiply by 3 so its a decent guess and you could verify for the next value if really necessary.
Reply 2
Original post by mqb2766
There is a hint in i) so
1/2(u_4 - 1) = 3^3
and u_2 and u_3 follow the same pattern and youve worked out their values. Also the term to term rule is roughly multiply by 3 so its a decent guess and you could verify for the next value if really necessary.

ohhh i kinda see it now 20240407_005712.jpg and yeah now its pretty obvious its n minus 1. I appreciate the help
(edited 2 months ago)
Reply 3
Original post by 43Explosion
ohhh i kinda see it now 20240407_005712.jpg and yeah now its pretty obvious its n minus 1. I appreciate the help

Note you could have used the term to term expression (subtract 1 from both sides) to get
u_(n+1) - 1 = 3(u_n - 1)
So a standard geometric sequence in z_n = u_n - 1. So starting from (u_1-1)=2, you get
u_(n+1) - 1 = 2*3^n
which is the form they want.

So even if you missed the "hence" hint, you could transform it into a standard geometric sequence to get the position to term expression. However, the proof by induction would be kind of redundant as you would have deduced the general expression. So the question is about "guessing" the expression and then proving it by induction.
(edited 2 months ago)

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