Question

Two A.P's have the same common difference. The difference between their ${100}^{th}$ terms is $111222333$. What is the difference between their Millionth terms?

Answer 1

Let the two A.P's be $a_1,a_2,a_3,....,a_n,....$ and $b_1,b_2,b_3,....b_n,....$

Also let the two common difference of two A.P's. Then,

$a_n=a_1(n-1)d$ and $b_n=b_1+(n-1)d$
$⟹a_n-b_n=\{a_1+(n-1\left)d\right\}-\{b_1+(n-1\left)d\right\}$
$⟹a_n-b_n=a_1-b_1$
Clearly, $a_n-b_n$ is independent of $n$ and is equal to $a_1-b_1$. In other words

$a_n-b_n=a_1-b_1$ for all $n\in N$.
$⟹a_{100}-b_{100}=a_1-b_1$

and, $a_k-b_k=a_1-b_1$, where $k=10,00,000$
But, $a_{100}-b_{100}=111222333$

$\therefore a_1-b_1=111222333$
$⟹a_k-b_k=a_1-b_1=111222333$, where $k=10,00,000$.
Hence, the difference between millionth terms is same as the difference between ${100}^{th}$ terms i.e., $111222333$.