Chapter 8.1, Problem 42E

a.

To determine

To Find : The first six terms of the sequence.

Answer to Problem 42E

The first six terms of the sequence are : 0, 0.25, 0.75, 1.5, 2.5, 3.8 .

Explanation of Solution

Given Information :

The n^th term of the sequence is given by, ${\text{I}}_{\text{n}}=100\left(\frac{{1.0025}^{\text{n}}-1}{0.0025}-\text{n}\right)$.............(1)

Calculation:

n^th term of the sequence is obtained by substituting the value of n in equation (1)

n = 1, gives the first term of the sequence, that is

  ${\text{I}}_1=100\left(\frac{{1.0025}^1-1}{0.0025}-1\right)=0$

n = 2, gives the second term of the sequence, that is

  ${\text{I}}_2=100\left(\frac{{1.0025}^2-1}{0.0025}-2\right)=0.25$

n = 3, gives the third term of the sequence, that is

  ${\text{I}}_3=100\left(\frac{{1.0025}^3-1}{0.0025}-3\right)=0.75$

n = 4, gives the fourth term of the sequence, that is

  ${\text{I}}_4=100\left(\frac{{1.0025}^4-1}{0.0025}-4\right)=1.5$

n = 5, gives the fifth term of the sequence, that is

  ${\text{I}}_5=100\left(\frac{{1.0025}^5-1}{0.0025}-5\right)=2.5$

n = 6, gives the sixth term of the sequence, that is

  ${\text{I}}_6=100\left(\frac{{1.0025}^6-1}{0.0025}-6\right)=3.8$

Conclusion:

Thus, the first six terms of the sequence are 0, 0.25, 0.75, 1.5, 2.5, 3.8 .

b.

To determine

To Find : The compound interest at the end of 2 years.

Explanation of Solution

Given Information:

$ 100 is deposited with an interest of 3% per annum and it is compounded monthly.

Calculation :

Equation for compound interest is, $\text{C}\text{.I=P}{\left(\text{1+}\frac{\text{r}}{\text{n}}\right)}^{\text{nt}}-\text{P}$

Where, C.I is the compound interest, P is the principal amount, r is the rate of interest, t the number of years, n the number of months as it’s compounded monthly.

According to the given informations, P = $100, r = 3% = 0.03, n = 12 and t = 2 .

  $\text{C}\text{.I=100}{\left(\text{1+}\frac{0.03}{12}\right)}^{\text{12}\times 2}-100$

  $=106.2-100=6.2$

Conclusion:

Thus,

The compound interest at the end of 2 years is $ 6.2 .