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Proof, EXT1 EQ-Bank 20

Use mathematical induction to prove that  \(3^n+7^n\)  is divisible by 10 for all odd \(n\).   (3 marks)

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\(\text{Proof (See Worked Solutions)}\)

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\(\text{Prove \(3^n+7^n\) is divisible by 10 for \(n\) odd.}\)

\(\text{Prove true for}\ \ n=1:\)

\(3^1+7^1=10 \ \ \text {(which is divisible by 10)}\)

\(\therefore \ \text{True for} \ \ n=1\)
 

\(\text{Assume true for} \ \ n=k\) :

\(3^k+7^k=10 P \ \text{(where \(P\) is an integer)}\)

\(\Rightarrow 3^k=10 P-7^k\ \ldots\ (1)\)
 

\(\text{Prove true for}\ \ n=k+2:\)

\(3^{k+2}+7^{k+2}\) \(=9 \times 3^k+49 \times 7^k\)
  \(=9\left(10 P-7^k\right)+49 \times 7^k\)
  \(=90 P-9 \times 7^k+49 \times 7^k\)
  \(=90P+40 \times 7^k\)
  \(=10\left(9 P + 4 \times 7^k\right)\)

 

\(\Rightarrow \ \text{True for} \ \ n=k+2\)

\(\therefore \ \text{Since true for \(n=1\), by PMI, true for integers \(n\) odd}.\)

Proof, EXT1 P1 2022 HSC 12f

Use mathematical induction to prove that  `15 ^(n)+6^(2n+1)`  is divisible by 7 for all integers  `n >= 0`.   (3 marks)

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`text{Proof (See Worked Solutions)}`

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`text{Prove}\ \ 15 ^(n)+6^(2n+1)\ \ text{is divisible by 7 for}\ \ n>=0`

`text{If}\ \ n=0:`

`15^0+6^1=7`

`:.\ text{True for}\ \ n=0`
 

`text{Assume true for}\ \ n=k`

`text{i.e.}\ \ 15^(k)+6^(2k+1)=7P\ \ text{(where}\ P\ text{is an integer)}`

`=>6^(2k+1)=7P-15^(k)\ \ …\ (1)`
 

`text{Prove true for}\ \ n=k+1`

`15^(k+1)+6^(2k+3)` `=15*15^(k)+6^2*6^(2k+1)`
  `=15*15^(k)+36(7P-15^(k))\ \ text{(see (1))}`
  `=15*15^(k)+36*7P-36*15^(k)`
  `=36*7P-21*15^(k)`
  `=7(36P-3*15^(k))\ \ text{(which is divisible by 7)}`

 
`=>\ text{True for}\ \ n=k+1`

`:.\ text{S}text{ince true for}\ \ n=0, text{by PMI, true for integers}\ \ n>=0`

Proof, EXT2 P1 EQ-Bank 28

Use mathematical induction to prove that  `7^n + 15^n`  is divisible by 11 where `n` is an odd integer.   (3 marks)

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`text(Proof)\ \ text{(See Worked Solutions)}`

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`text(Prove)\ \ 7^n + 15^n\ \ text(is divisible by 11, where)\ n\ text(is odd)`

`text(If)\ \ n = 1:`

`7^n + 15^n` `= 7 + 15 = 22`
`22/11` `= 2\ \ \ text{(Divisible by 11)}`

 
`text(Assume true for)\ \ n = k:`

`text(i.e.)\ 7^k + 15^k` `= 11P\ \ \ \ text{(}P\ text(integer) text{)}`
`7^k` `= 11P\ – 15^k\ \ \ \ \ …\ text{(∗)}`

 
`text(Prove true for)\ \ n = k + 2:`

`7^(k + 2) + 15^(k + 2)` `= 7^2 * 7^k + 15^2 * 15^k`
  `= 49 (11P-15^k) + 225 (15^k)\ \ \ text(… from)\ text{(∗)}`
  `= 49 * 11P-49 * 15^k + 225 (15^k)`
  `= 49 * 11P + 176 * 15^k`
  `= 11 (49P + 16 * 15^k)`

  
`=>text(True for)\ n = k + 2`

`:.\ text(S)text(ince true for)\ n = 1,\ text(by PMI, true for odd integral)\ \ n >= 1`