BdMO National Higher Secondary 2014/4

[samiul_samin]

$97+98+99+.........+114+115=2014$.Here sum of $19$ consequative numbers is $2014$.Find the largest number of consequative positive integers whose sum is exactly $2014$ and justify why you think this must be the largest number.

[samiul_samin]

Hint

Use Sequence and Factorization.

Answer

$\fbox {53}$

[samiul_samin]

Solution

Suppose, $a+a+d+a+2d+.........+a+nd=2014$
So, $\dfrac n2 ${$2a+(n-1)d$}$=2014\Rightarrow n(2a+nd-d)=4028$...$(1)$
Now, $4028=1\times 4028=2\times 2014=19\times 212=53\times 76$
The gretest $n=53$ if we put $n $ in $(1)$.
If $n=53$,$a=12,d=1$
(Just calculate to ensure the answer).
The desired largest number is $\fbox {53}$
[Justified]

[samiul_samin]

Can anyone give a different solution?