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JEE Advanced2023MathematicsSequences and SeriesActual

Let 7 5 … 5 ⏞ 7 r denote the r + 2 digit number where the first and the last digits are 7 and the remaining r digits are 5 . Consider the sum S = 77 + 757 + 7557 + … + 7 5 … 5 ⏞ 7 98 . If S = 7 5 … 5 ⏞ 99 7 + m n , where m and n are natural numbers less than 3000 , then the value of m + n is

Correct answer

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Step-by-step solution

Given, S = 77 + 757 + 7557 + … + 7 5 … 5 ⏞ 7 98 ⇒ S = 7 × 10 + 7 + 7 × 100 + 5 × 10 + 7 + 7 × 1000 + 5 × 100 + 5 × 10 + 7 … + 7 5 … 5 ⏞ 7 98 ⇒ S = 7 10 + 10 2 + … + 10 99 + 50 1 + 11 + … + 111 … 1 ⏞ 98 + 7 × 99 ⇒ S = 70 10 99 - 1 9 + 50 9 10 - 1 + 10 2 - 1 + … + 10 98 - 1 + 7 × 99 ⇒ S = 70 10 99 - 1 9 + 50 9 10 10 98 - 1 9 - 98 + 7 × 99 ⇒ S = 7 × 10 100 9 - 70 9 + 50 9 10

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