Question

A sum of Rs $1000$ is invested at $8\%$ simple interest per year. Calculate the interest at the end of each year. Do these interests form an AP? If so, find the interest at the end of $30$ years making use of this fact.

Answer 1

Step 1. Finding the interest at the end of the first year:

The formula for simple interest is$I=PRT$, where $P=$ principle, $R=$rate of interest, $T=$ time ( in year)

Here $P=1000,R=8\%,T=1$, $I_1=$Interest at the end of the first year

$I_1=1000\times 8\%\times 1$

$$\begin{gathered} =1000\times \frac{8}{100}\times 1 \\ =80 \end{gathered}$$

Step 2. Finding the interest at the end of the second year:

The formula for simple interest is$I=PRT$,

Here $P=1000,R=8\%,T=2$,$I_2=$Interest at the end of the second year

$I_2=1000\times 8\%\times 2$

$$\begin{gathered} =1000\times \frac{8}{100}\times 2 \\ =160 \end{gathered}$$

Step 3. Finding the interest at the end of the third year:

The formula for simple interest is$I=PRT$,

Here $P=1000,R=8\%,T=3$,$I_3=$Interest at the end of the third year

$$\begin{gathered} I_3=1000\times 8\%\times 3 \\ =1000\times \frac{8}{100}\times 3 \\ =240 \end{gathered}$$

Step 4. Finding the interest at the end of the fourth year:

The formula for simple interest is$I=PRT$,

Here $P=1000,R=8\%,T=4$,$I_3=$Interest at the end of the fourth year

$$\begin{gathered} I_4=1000\times 8\%\times 4 \\ =1000\times \frac{8}{100}\times 4 \\ =320 \end{gathered}$$

Step 5: Check whether the interests are in A.P

The sequence is $80,160,240,320\dots .$

If the sequence is an AP then the common difference will be the same.

Here

First-term $t_1=80$

Second term $t_2=160$

Third term $t_3=240$

Fourth term $t_4=320$

The formula of common difference is$d=n^{th}term-{(n-1)}^{th}term$

Here

$$\begin{gathered} t_2-t_1=160-80=80 \\ {t}_3-t_2=240-160=80 \\ {t}_4-t_3=320-160=80 \end{gathered}$$

Since $t_2-t_1=t_3-t_2=t_4-t_3=80$

Thus it is an A.P.

Step 6. Finding the interest at end of $30$ years

The formula of the $n^{th}$ term of a sequence is $t_n=a+(n-1)d$, where $a$ is the first term of A.P. and $d$ is the common difference of A.P.

Here

$$\begin{gathered} a=\mathrm{First}\mathrm{term}=80 \\ n=30 \\ d=\mathrm{common}\mathrm{difference}=80 \end{gathered}$$

The interest at the end of $30$ years is $80+(30-1)80$

$=80+2320$

$=2400$.

Hence, the interest at the end of each year is $80,160,240,320,....$ These interests are from an AP. The interest at the end of $30$ years is Rs. $2400$.