Prove by Induction!

Another doubt in induction, please help me out.

Show that 27n+35n52\cdot7^{n} + 3\cdot5^{n} -5 is divisible by 24.

Note by Swapnil Das
4 years ago

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Step 11: when n=1n=1, we have 27n+35n5=27+355=242 \cdot 7^{n}+3 \cdot 5^{n}-5=2 \cdot 7+3 \cdot 5 -5=24 which is divisible by 2424.

Step 22: Assume this to be true for n=mn=m. Then 27m+35m52 \cdot 7^{m}+3 \cdot 5^{m}-5 is divisible by 2424.

Step 33: Now, 27m+1+35m+152 \cdot 7^{m+1}+3 \cdot 5^{m+1}-5

=27m×7+35m×55=2 \cdot 7^{m} \times 7+ 3 \cdot 5^{m} \times 5-5

=147m+155m5=14 \cdot 7^{m}+15 \cdot 5^{m}-5

=27m+35m5+127m+125m=2 \cdot 7^{m}+3 \cdot 5^{m}-5+12 \cdot 7^{m}+12 \cdot 5^{m}

Since 27m+35m52 \cdot 7^{m}+3 \cdot 5^{m}-5 is divisible by 2424, we have only to prove that 127m+125m12 \cdot 7^{m}+12 \cdot 5^{m} is divisible by 2424.

127m+125m=12(7m+5m)12 \cdot 7^{m}+12 \cdot 5^{m}=12(7^{m}+5^{m}) which is divisible by 2424 since 7m7^{m} and 5m5^{m} are both odd numbers and their sum is an even number.

Hence 27n+35n52 \cdot 7^{n}+3 \cdot 5^{n}-5 is divisible by 2424 for all integers n>0n>0.

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Excellent, thank you genius!

Swapnil Das - 4 years ago

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Your Welcome :)

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Hint: Show that it is true for n=1. Then show that if the statement is true for any arbitrary positive integer m, then it is true for m+1.

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That's an obvious hint lol xD

Nihar Mahajan - 4 years ago

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Can you show the m+1 step? This is where I'm getting stuck at.

Swapnil Das - 4 years ago

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Only tool (with me ):MATHEMATICAL INDUCTION

Aakash Khandelwal - 4 years ago

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I'm surprised that you don't know Induction. You should read the wiki and be familiar with it. This is a standard problem in divisibility by induction.

Calvin Lin Staff - 4 years ago

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Lol, I assumed powers of 55 to be even by mistake, and thus couldn't get the result!

Swapnil Das - 4 years ago

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This is true only for n equals 1. It is not true for numbers bigger than 1.

mohamed sultan - 3 years, 11 months ago

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@ Mohamed Sultan - You are wrong. Go back and try it for n = 2, 3, and 4, for instance.

Linda Slovik - 1 year, 2 months ago

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