JEE Advanced · Mathematics · 5. Sequences & Series

Let $\overset{r}{7 \overset{⏞}{5 \dots 5} 7}$ 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 $S = 77 + 757 + 7557 + \dots + \overset{98}{7 \overset{⏞}{5 \dots 5} 7}$ . If $S = \frac{7 \overset{99}{\overset{⏞}{5 \dots 5}} 7 + m}{n}$ , where $m$ and $n$ are natural numbers less than $3000$ , then the value of $m + n$ is

A 1219
B 1200
C 1554
D 1624

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Answer & Solution

Correct Answer

(A) 1219

Step-by-step Solution

Detailed explanation

Given,
$S=77+757+7557+\ldots+7 \overbrace{5 \ldots 57}^{98}$
$\Rightarrow S=7 \times 10+7+7 \times 100+5 \times 10+7$ $+~7 \times 1000+5 \times 100+5 \times 10+7 \ldots+$ $7 \overbrace{5 \ldots 5}^{98} $
$ \Rightarrow S=7\left(10+10^2+\ldots+10^{99}\right)+50$ $(1+11+\ldots+\overbrace{111 \ldots 1}^{98})+7 \times 99 $
$ \Rightarrow S=70\left(\frac{10^{99}-1}{9}\right)+\frac{50}{9}[(10-1)+\left(10^2-1\right)$ $+\ldots+\left(10^{98}-1\right)]+7 \times 99 $
$ \Rightarrow S=70\left(\frac{10^{99}-1}{9}\right)+\frac{50}{9}\left[10\left(\frac{10^{98}-1}{9}\right)-98\right]$ $+~7 \times 99 $
$ \Rightarrow S=\frac{7 \times 10^{100}}{9}-\frac{70}{9}+\frac{50}{9}\left[\frac{10^{99}-1-9}{9}-98\right]$ $+~7 \times 99 $
$ \Rightarrow S=\frac{7 \times 10^{100}}{9}-\frac{70}{9}+\frac{50}{9}[\overbrace{111 \ldots 1}^{99}-99]$ $+~7 \times 99 $
$ \Rightarrow S=\frac{7 \times 10^{100}-70+\overbrace{555 \ldots 50}^{99}}{9}-550+693$
$\Rightarrow S=\frac{7 \overbrace{555 \ldots .5}^{99}-70+143 \times 9}{9}$
$\Rightarrow S=\frac{7 \overbrace{55 \ldots 5}^{99} 7+1210}{9}$
So, on comparing we get, $m+n=1210+9=1219$

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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$P$	$t_{2 g}^{6} e_{g}^{0}$	$1$	${[Fe {(H_{2} O)}_{6}]}^{2 +}$
$Q$	$t_{2 g}^{3} e_{g}^{2}$	$2$	${[Mn {(H_{2} O)}_{6}]}^{2 +}$
$R$	$e^{2} t_{2}^{3}$	$3$	${[Co {({NH}_{3})}_{6}]}^{3 +}$
$S$	$t_{2 g}^{4} e_{g}^{2}$	$4$	${[{FeCl}_{4}]}^{-}$
		$5$	${[{CoCl}_{4}]}^{2 -}$

Let 75…5⏞7r 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…5⏞798. If S= 75…5 ⏞997+mn, where m and n are natural numbers less than 3000, then the value of m+n is

Step-by-step Solution