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
A. A
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 99 - 1 - 9 9 - 98 + 7 × 99 ⇒ S = 7 × 10 100 9 - 70 9 + 50 9 111 … 1 ⏞ 99 - 99 + 7 × 99 ⇒ S = 7 × 10 100 - 70 + 555 … 5 ⏞ 99 0 9 - 550 + 693 ⇒ S = 7 555 … . 5 ⏞ 99 - 70 + 143 × 9 9 ⇒ S = 7 55 … 5 ⏞ 99 7 + 1210 9 So, on comparing we get, m + n = 1210 + 9 = 1219
Related: Mathematics — Sequences and Series · All PYQ Banks