Question

Show tha 7+77+777+7777..... Principle of mathematical induction

Answer 1

The complete question is:

Using Principal of Induction, prove

7 + 77 + 777 + ... + 7777...77 (n digits)

= 7/81 * (10ⁿ⁺¹ - 9n - 10)

Proof:

Step 1.

For n = 1 the statement is true because

7 = 7/81 * (10¹⁺¹ - 9.1 - 10)

Step 2.

Let us assume that the statement is true for some natural number n = k.

Then 7 + 77 + 777 + ... + 777...77 (k digits)

= 7/81 * (10ᵏ⁺¹ - 9k - 10)

Now for n = k + 1, we get

7 + 77 + 777 + ... + 777...777 + 7777...777 (k + 1 digits)

= (7 + 77 + 777 + ... + 777.777) (k digits) + 7777...7777 (k + 1) digits

= 7/81 * (10ᵏ⁺¹ - 9k - 10) + 7 * 1111...1111 (k + 1) digits

= 7/81 * (10ᵏ⁺¹ - 9k - 10) + 7/9 * (10ᵏ⁺¹ - 1) (for k + 1 digits)

= 7/81 * (10ᵏ⁺¹ - 9k - 10 + 9 * 10ᵏ⁺¹ - 9)

= 7/81 * (10.10ᵏ⁺¹ - 9k - 9 - 10)

= 7/81 * {10¹⁺ᵏ⁺¹ - 9 (k + 1) - 10}

= 7/81 * {10ᵏ⁺¹⁺¹ - 9 (k + 1) - 10}

This shows that the statement is true for the natural number (k + 1) if it is true for k.

Therefore by the Principle of Induction, the statement is true for all natural numbers n.

This completes the proof.

Answer 2

Answer:

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