In which of the following situations, do the lists of numbers involved form an $ A P $ ? Give reasons for your answers.
The amount of money in the account of Varun at the end of every year when Rs 1000 is deposited at simple interest of $ 10 \% $ per annum.

Given:

The amount of money in the account of Varun at the end of every year when Rs 1000 is deposited at simple interest of \( 10 \% \) per annum.

To do:

We have to check whether  the list of numbers involved in the given situation forms an \( A P \).

Solution:

We know that,

Simple interest $=\frac{\text { Principal } \times \text { Rate } \times \text { Time }}{100}$

$=\frac{1000 \times 10 \times 1}{100}$

$=100$

This implies,

The amount of money in the account of Varun at the end of each year(in Rs.) is,

$1000, (1000+100 \times 1),(1000+100 \times 2),(1000+100 \times 3), \ldots $

The sequence formed is,

$1000,1100,1200,1300, \ldots $

In the given sequence,

$a_1=1000, a_2=1100, a_3=1200$

$a_2-a_1=1100-1000=100$ 

$a_3-a_2=1200-1100=100$

Here,

$a_3-a_2=a_2-a_1$

Therefore,

The list of numbers involved in the given situation forms an \( A P \).