Dackor
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I am studying under the WJEC Further Mathematics subject, and I am stumped with one Induction question and hope anyone can help.

The question is:

Show that:
f\left( n\right) =3^{2n+1}-2^{2n-1}

is divisible by 5, for all positive integral values of n.



Thanks in advance,

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waxwing
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What are you stuck on? For these kinds of proofs, you need to (a) check it's true for the lowest value (in this case n=1 ), then write down the expression replacing n \rightarrow n+1 , then try to figure out how you can prove that's true, given the original statement about n. Well that's the basic idea.
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Dackor
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Hi there,

Thanks for the quick reply
I believe I've figured it now though.

I have indeed proven the statement to be true for the case where n=1;

f\left( 1\right) =3^{2\left( 1\right) +1}-2^{2\left( 1\right)-1}
f\left( 1\right) =3^{3}-2^{1}
f\left( 1\right) =27-2
f\left( 1\right) =25

25 is divisible by 5, thus the result is true for the case where n=1.

I then assumed that the result obtained from the case where n=k to be true (Where k is an arbitrary value);

f\left( k\right) =3^{2k+1}-2^{2k-1}

I then considered the case where n=k+1;

f\left( k+1\right) =3^{2(k+1)+1}-2^{2(k+1)-1}
f\left( k+1\right) =3^{2k+3}-2^{2k+1}
f\left( k+1\right) =3^{2k}.3^{3}-2^{2k}.2^{1}
f\left( k+1\right) =3^{2k}.3^{2}.3^{1}-2^{2k}.2^{-1}.2^{2}
f\left( k+1\right) =3^{2}(3^{2k}.3^{1})-2^{2}(2^{2k}.2^{-1})
f\left( k+1\right) =9(3^{2k+1})-4(2^{2k-1})

Then, from this stage, I added the f\left( k\right) term;

f\left( k+1\right)+f\left( k\right) =10(3^{2k+1})-5(2^{2k-1})

And, from this result; Both 10 and 5 are divisible by 5;

Therefore, the sum of f\left( k+1\right) and f\left( k\right) is divisible by 5 \forall n\in \mathbb{Z} ^{+}.

Is this correct?

Thanks,
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waxwing
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Yep, you got it

Just make sure you lay out your proof clearly. You proved f(k+1)+f(k) is divisible by 5, just clarify that you're proving that if f(k) is divisible by 5, then f(k+1) is divisible by 5. Most examining boards will want you to use the concept of a 'hypothesis', and usually have a "standard" way of laying out the proof. Consult textbook/teacher for details.
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Dackor
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(Original post by waxwing)
Yep, you got it

Just make sure you lay out your proof clearly. You proved f(k+1)+f(k) is divisible by 5, just clarify that you're proving that if f(k) is divisible by 5, then f(k+1) is divisible by 5. Most examining boards will want you to use the concept of a 'hypothesis', and usually have a "standard" way of laying out the proof. Consult textbook/teacher for details.
Thanks so much for the help and tips
That's Proof by Induction done for me now! Just waiting on starting Complex Numbers and Transformation of Matrices now, which is quite daunting looking at in textbooks!
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