Question

Two APs have the same common difference. The difference between their $100th$ terms is $100.$ What is the difference between their $1000th$ terms ?

Answer 1

Having common difference 'd', the APs

1st set $a,a+d,a+2d$

2nd set $b,b+d,b+2d$

${100}^{th}$ term of $1^{st}$ set - ${100}^{th}$ term of $2^{nd}$ set $=100$

$\therefore a+99d-(b+99d)=100$

$a+99d-b-99d=100$

$a-b=100$

Similarly,

$1000th$ term of $1st$ set $=1+999d$

$1000th$ term of $2nd$ set $=b+999d$

Their difference

$=a+999d-(b+999d)$

$=a+999d-b-999d$

$=a-b=100.$