Let 75 . . . 57 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 + ? + 75 . . . 57 98 . If S = 75 . . . 57 99 + m n , where m and n are natur

Question

Let $\overset{r}{\overset{⏞}{75 . . . 57}}$ 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 $S = 77 + 757 + 7557 + \dots + \overset{98}{\overset{⏞}{75 . . . 57}} .$ If $S = \frac{\overset{99}{\overset{⏞}{75 . . . 57}} + m}{n},$ where $m$ and $n$ are natural numbers less than 3000, then the value of $m + n$ is [2023]

Smart Achievers · Accepted Answer

(1219)

$Given S = 77 + 757 + 7557 + \dots + \overset{98}{\overset{⏞}{75 . . . 57}}$

$\frac{10 S = 770 + 7570 + \dots + 75 . . . . . . . . 570 + \overset{98}{\overset{⏞}{75 . . . 570}}}{- 9 S = 77 - \underset{98 times}{\underset{⏟}{13 - 13 - \dots - 13}} - \underset{98}{\underset{⏟}{75 \dots 570}}}$

$9 S = - 77 + 13 \times 98 + \underset{99}{\underset{⏟}{75 \dots 57}} + 13$

$S = \frac{\overset{99}{\overset{⏞}{75 . . . . . 57}} + 1210}{9}$

$\Rightarrow m = 1210, n = 9 \Rightarrow m + n = 1219 .$

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Solution

Q. Let 75...57⏞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+⋯+75...57⏞98. If S=75...57⏞99+mn, where m and n are natural numbers less than 3000, then the value of m+n is [2023]

Ans. (1219) Given S=77+757+7557+⋯+75...57⏞98 10S=770+7570+⋯+75........570+75...570⏞98-9S= 77-13-13-⋯-13⏟98 times-75…570⏟98 9S=-77+13×98+75…57⏟99+13 S=75.....57⏞99+12109 ⇒ m=1210, n=9 ⇒ m+n=1219.

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