document.write( "Question 1205176: Prove by mathematical induction that 8𝑛 − 3𝑛
\n" ); document.write( "is divisible by 5 for all positive integers n \n" ); document.write( "
Algebra.Com's Answer #841817 by math_tutor2020(3817)\"\" \"About 
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\n" ); document.write( "\"8n+-+3n+=+%288-3%29n+=+5n\" is clearly a multiple of 5 when n is an integer.\r
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\n" ); document.write( "\n" ); document.write( "Eg:
\n" ); document.write( "n = 1 leads to 5n = 5*1 = 5
\n" ); document.write( "n = 2 leads to 5n = 5*2 = 10
\n" ); document.write( "n = 3 leads to 5n = 5*3 = 15\r
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\n" ); document.write( "\n" ); document.write( "If you meant to say \"8%5En+-+3%5En\", then we can do a proof by induction.\r
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\n" ); document.write( "\n" ); document.write( "Base case: n = 1
\n" ); document.write( "\"8%5En+-+3%5En=8%5E1+-+3%5E1+=+8+-+3+=+5\"
\n" ); document.write( "This proves that \"8%5En+-+3%5En\" is a multiple of 5 when n = 1.
\n" ); document.write( "The base case is done.\r
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\n" ); document.write( "\n" ); document.write( "Inductive Step:
\n" ); document.write( "Assume that \"8%5Ek+-+3%5Ek\" is a multiple of 5 for some integer k > 1.
\n" ); document.write( "That will mean \"8%5Ek+-+3%5Ek+=+5m\" for some integer m.
\n" ); document.write( "Let's say we isolated the 8^k portion
\n" ); document.write( "\"8%5Ek+=+5m+%2B+3%5Ek\"
\n" ); document.write( "which will be useful in a substitution step later.\r
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\n" ); document.write( "\n" ); document.write( "The goal is to show that \"8%5E%28k%2B1%29+-+3%5E%28k%2B1%29\" is also a multiple of 5 based on the assumption above.
\n" ); document.write( "This will produce a domino effect to prove \"8%5En+-+3%5En\" is a multiple of 5 for any integer \"n+%3E=+1\"\r
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\n" ); document.write( "\n" ); document.write( "\"8%5E%28k%2B1%29+-+3%5E%28k%2B1%29\"\r
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\n" ); document.write( "\n" ); document.write( "\"8%5Ek%2A8%5E1+-+3%5Ek%2A3%5E1\"\r
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\n" ); document.write( "\n" ); document.write( "\"8%2A8%5Ek+-+3%2A3%5Ek\"\r
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\n" ); document.write( "\n" ); document.write( "\"8%2A%285m+%2B+3%5Ek%29+-+3%2A3%5Ek\" Substitution step. Replace 8^k with 5m+3^k.\r
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\n" ); document.write( "\n" ); document.write( "\"8%2A5m+%2B+8%2A3%5Ek+-+3%2A3%5Ek\"\r
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\n" ); document.write( "\"5%2A8m+%2B+%288-3%29%2A3%5Ek\"\r
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\n" ); document.write( "\"5%2A8m+%2B+5%2A3%5Ek\"\r
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\n" ); document.write( "\"5%2A%288m+%2B+3%5Ek%29\"\r
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\n" ); document.write( "\"5%2A%28matrix%281%2C2%2C%22some%22%2C%22integer%22%29%29\"
\n" ); document.write( "This proves that if \"8%5Ek+-+3%5Ek\" is a multiple of 5, then \"8%5E%28k%2B1%29+-+3%5E%28k%2B1%29\" is also a multiple of 5.
\n" ); document.write( "The inductive step is done.\r
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\n" ); document.write( "\n" ); document.write( "This wraps up the induction proof.\r
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\n" ); document.write( "\n" ); document.write( "More practice with induction proofs
\n" ); document.write( "https://www.algebra.com/algebra/homework/Sequences-and-series/Sequences-and-series.faq.question.1203186.html\r
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\n" ); document.write( "\n" ); document.write( "Here's another approach to proving that \"8%5En+-+3%5En\" is a multiple of 5 when n is an integer and \"n+%3E=+1\"\r
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\n" ); document.write( "\n" ); document.write( "Let's look at the powers of 8
\n" ); document.write( "8^1 = 8
\n" ); document.write( "8^2 = 64
\n" ); document.write( "8^3 = 512
\n" ); document.write( "8^4 = 4096
\n" ); document.write( "8^5 = 32768
\n" ); document.write( "The units digits from top to bottom are: 8, 4, 2, 6, 8
\n" ); document.write( "Once we arrive at 8 again, the cycle repeats.
\n" ); document.write( "Therefore, those are the only units digits possible.
\n" ); document.write( "Note the cycle is 4 items long. \r
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\n" ); document.write( "\n" ); document.write( "Let's look at the powers of 3
\n" ); document.write( "3^1 = 3
\n" ); document.write( "3^2 = 9
\n" ); document.write( "3^3 = 27
\n" ); document.write( "3^4 = 81
\n" ); document.write( "3^5 = 243
\n" ); document.write( "The units digits are 3, 9, 7, 1, 3
\n" ); document.write( "After arriving at 3 again, the cycle repeats.
\n" ); document.write( "Those are the only units digits possible.
\n" ); document.write( "Note the cycle is 4 items long, the exact same length as the previous cycle. \r
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\n" ); document.write( "\n" ); document.write( "Let's arrange that info into a table.
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n8^n3^nUnits digit of 8^nUnits digit of 3^n
18383
264949
35122727
440968161
53276824383

\n" ); document.write( "When n = 1 the units digits for powers of 8 and 3 are 8 and 3 respectively.
\n" ); document.write( "8-3 = 5 shows \"8%5En+-+3%5En\" is a multiple of 5 when n = 1.\r
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\n" ); document.write( "\n" ); document.write( "When n = 2 the units digits for powers of 8 and 3 are 4 and 9 respectively.
\n" ); document.write( "Think of the 4 as 14 since we can borrow a ten.
\n" ); document.write( "Afterward 14-9 = 5 shows \"8%5En+-+3%5En\" is a multiple of 5 when n = 2.
\n" ); document.write( "We don't have to look at the entire difference. All we need is the difference of the units digits.
\n" ); document.write( "Recall that a number is a multiple of 5 if it ends with 0 or 5. \r
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\n" ); document.write( "\n" ); document.write( "When n = 3 the units digits for powers of 8 and 3 are 2 and 7 respectively.
\n" ); document.write( "Think of the 2 as 12 since we can borrow a ten.
\n" ); document.write( "Afterward 12-7 = 5 shows \"8%5En+-+3%5En\" is a multiple of 5 when n = 3.\r
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\n" ); document.write( "\n" ); document.write( "When n = 4 the units digits for powers of 8 and 3 are 6 and 1 respectively.
\n" ); document.write( "Subtract the units digits 6-1 = 5 to show \"8%5En+-+3%5En\" is a multiple of 5 when n = 4.\r
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\n" ); document.write( "\n" ); document.write( "For n = 5, it's a repeat of n = 1
\n" ); document.write( "n = 6 is a repeat of n = 2
\n" ); document.write( "n = 7 is a repeat of n = 3
\n" ); document.write( "and so on.
\n" ); document.write( "This concludes an alternate proof where we avoid induction. \r
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\n" ); document.write( "\n" ); document.write( "Despite the fact this alternative is available, I recommend getting to know induction more because it shows up a lot in math proofs.
\n" ); document.write( "Also, it's a good idea to follow the teachers/textbooks instructions to get full marks.
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