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Anonymous asked in Science & MathematicsEngineering · 9 years ago

Good in mathematical Induction?

Prove statement by Mathematical Induction

7 + 77 + 777 + ... + 777...7(n digits) = 7(10^n+1 - 9n - 10) / 81

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  • 9 years ago
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    Also means 1+11+111+...+ n1's= 10^n+1-9n-10/81

    consider n-1's 1+11+111+...+ (n-1)1's = 10^n-9(n-1)-10/81. You get this when you replace n by n-1

    Subtract to get n(1)'s = (10^n+1-10^n-9)/81

    If n=5, LHS = 11111. RHS = (10^6-10^5-9)/81, which is indeed true. 10^6-10^5 = 9 followed by 5 zeroes. Subtract 9 and then divide by 81, and you get 11111.

    LHS = n(1)'s. Multiply by 81, and you get 8 followed by (n-1)9's followed by a 1. Add 9 to this, and you get 9 followed by (n-1) 0's.

    RHS numerator is 1 followed by n(0's) minus 1 followed by n-1(0)'s = 9 followed by (n-1)(0)'s.

    So RHS equal LHS

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