Question:
Split 207 into three parts such that these are in AP and the product of the two smaller parts is 4623.
Solution:
Let the three parts of the number 207 are (a – d), a and (a + d), which are in AP
Now, by given condition,
$\Rightarrow \quad$ Sum of these parts $=207$
$\Rightarrow \quad a-d+a+a+d=207$
$\Rightarrow \quad 3 a=207$
$a=69$
Given that, product of the two smaller parts $=4623$
$\Rightarrow \quad a(a-d)=4623$
$\Rightarrow \quad 69 \cdot(69-d)=4623$
$\Rightarrow \quad 69-d=67$
$\Rightarrow \quad \quad d=69-67=2$
So, $\quad$ first part $=a-d=69-2=67$,
second part $=a=69$
and $\quad$ third part $=a+d=69+2=71$,
Hence, required three parts are 67, 69, 71.