InterviewSolution
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InterviewSolution
| 1. |
`3^(2n+2)-8n-9` divisible by `8` |
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Answer» Let `P(n) =3^(2n+2) -8n-9` for n=1 `p(1) =3^(4)-8(1) -9 =81 -17 =64 =8 (8)` Which is divisible by 8 `rArrP (n)` be true for n=1 Let P (n) be true for n=K. `:.P(k) : 3^(2K+2) -8k-9=8lambda (" say ")` `" Where " lambda in I` for n=k+1 `P(k+1):3^(2(k+1)+2) -8(k+1)-9` `=3^(2).3^(2k+2) -8k-8-9` `=9[8lambda+8k+9]-8k-17` `=9.8lambda+72k+81-8K-17` `=9.8lambda+64lambda+64` `=8[9lambda+8k+8]` which is divisible by 8 `rArr` P (n) is also true for n=K+1 Hence from the principle of mathematical induction p(n) is true for all natural numbes n. |
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