Question

In which of the following situations, the sequence formed will form an A.P.?
(i) Number of students left in the school auditorium from the total strength of $1000$ students when they leave the auditorium in batches of $25$.
(ii) The amount of money in the account every year when Rs. $100$ are deposited annually to accumulate at compound interest at $4\%$ per annum.

Answer 1

(i) We have,
Total strength of students in the auditorium $=1000$

Number of students left in the auditorium when first batch of $25$ students leave the auditorium $=1000-25=975$
Number of students left in the auditorium when second batch of $25$ students leave the auditorium $=975-25=950$
Number of students left in the auditorium when third batch of $25$ students leave the auditorium $=950-25=925$ and so on.

Thus, the number of students left in the auditorium at different stages are
$1000,975,950,925,....$
Clearly, it is an A.P. with first term $1000$ and a common difference $-25$.
(ii) We know that if $P$ is the principal and $r$ per annum is the rate of interest, compound annually,

then the amount $A_n$ at the end of $n$ years is given by
$A_n=P{(1+\frac{r}{100})}^n$
Here $P=Rs100$ and $r=4$.

$\therefore A_n=100{(1+\frac{4}{100})}^n=100\times {\left(\frac{26}{25}\right)}^n=100\times (1.04{)}^n$

Thus, the amount of monkey in the account at the end of different years is given by
$Rs100\times 1.04,Rs100\times (1.04{)}^2,Rs100\times (1.04{)}^3,....$
or $Rs104,Rs108.16,Rs112.48,...$ Clearly, it is not forming an A.P. as common difference is not same.