Question

The sum of four numbers is $80$. If we subtract $1,2,11,44$ from these numbers in that order. We obtain an arithmetic progression. Find the numbers.

Answer 1

Let

$a+b+c+d=80$

$\Rightarrow (a-1),(b-2),(c-11),(d-44)$ are in AP

So,

$2(b-2)=(c-11)+(a-1)$

$2b=a+c-8$...(1)

$2(c-11)=(d-44)+(b-2)$

$2c=b+d-24$...(2)

Adding (1) and (2), we get,

$b+c=24\Rightarrow a+d=80-24=56$

Let $c=18,b=6$

so, $b-2=4,c-11=7$

$\Rightarrow d=7-4=3$

so, first term = 1

and last term =10

$\Rightarrow a=1+1=2,a+d=56\Rightarrow d=54$

Therefore the numbers are

$a=2$

$b=6$

$c=18$

$d=54$