Interest

Examples of Interest in the following topics:

• Interest Compounded Continuously

  • Compound interest is accrued when interest is earned not only on principal, but on previously accrued interest: it is interest on interest.
  • In simple interest, interest is accrued on the principal alone.
  • In compound interest, interest is accrued on both the principal and on prior interest earned.
  • That is because the difference is that compound interest earns interest on both the principal and prior interest.
  • You earn the most interest when interest is compounded continuously.

• Formulas and Problem-Solving

  • For example, one can use a linear equation to determine the amount of interest accrued on a home equity line of credit after a given amount of time.
  • Consider the hypothetical situation in which you need money to make home improvements and can open a $20,000 credit line at an interest rate of 2.5% per year.
  • Where I is interest, p is the principal amount loaned ($20,000), r is the interest rate (2%, or 0.02) per year, and T is the number of years elapsed (15 months will be 1.25 years).
  • Plugging the known values into the above formula, we can determine that you will pay $500 in interest.

• The Number e

  • One of the many places the number $e$ plays a role in mathematics is in the formula for compound interest.
  • Jacob Bernoulli discovered this constant by asking questions related to the amount of money in an account after a certain number of years, if the interest is compounded $n$ times per year.
  • He was able to come up with the formula that if the interest rate is $r$ percent and is calculated $n$ times per year, and the account originally contained $P$ dollars, then the amount in the account after $t$ years is given by $A=P(1+{r \over n})^{nt}.$ By then asking about what happens as $n$ gets arbitrarily large, he was able to come up with the formula for continuously compounded interest, which is $A=Pe^{rt}

• Graphs of Exponential Functions, Base e

  • The function $f(x) = e^x$ is a basic exponential function with some very interesting properties.

• Factors

  • Prime factorization is a particular type of factorization that breaks a number of interest into prime numbers that when multiplied back together produce the original number.
  • In a factor tree, the number of interest is written at the top.

• Introduction to Exponents

  • Exponentiation is used frequently in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.

• Problem-Solving

  • Compound interest at a constant interest rate provides exponential growth of the capital.

• Geometric Sequences

  • An interesting result of the definition of a geometric progression is that for any value of the common ratio, any three consecutive terms $a$, $b$, and $c$ will satisfy the following equation:

• Logarithmic Functions

  • The irrational number  $e\approx 2.718$ arises naturally in financial mathematics, in computations having to do with compound interest and annuities.

• The Leading-Term Test

  • (Notice that we do not care about $x = 0$ since we are only interested in very large $x.$)