Question

If two consecutive numbers are removed from $1,2,3,\dots ,50$, the arithmetic mean becomes $\frac{105}{4}$, then the removed number are

• A. $7$ and $8$
• B. $8$ and $9$
• C. $9$ and $10$
• D. $10$ and $11$

Answer 1

The correct option is A $7$ and $8$

Given A.P. is $1,2,3,\dots ,50$
Sum of $50$ terms
$S_{50}=\frac{50}{2}[1+50]\Rightarrow S_n=1275$

Let the removed numbers be $p$ and $(p+1)$
From given condition,
$\frac{1275-(p+p+1)}{48}=\frac{105}{4}\Rightarrow 1274-2p=12\times 105\Rightarrow p=7$

Hence, the two consecutive number removed are $7$ and $8$.