Applied Business Mathematics

Defines linear equations in one variable and explains degree 1 and the standard form x + b = 0. Demonstrates solving and roots with examples like 3x+5=0 and y=2x+3.

Learn the four rules for solving a linear equation in one variable: add, subtract, multiply, and divide by equal numbers, plus transposition to shift terms and signs.

Learn to solve linear equations in one variable using formal rules and the transposition shortcut, isolating the variable across multiple examples and presenting the solution set.

Introduces linear equations with two or more variables and defines the standard form ax + by + c = 0, with methods to solve by fixing a variable.

Apply linear equations with two or more variables to real-world problems, such as production of models A and B under labor constraints with 5x+6y=240, and budget-based juice blends with 2x1+4x2+5x3=6000.

Discover how sequences operate in business, finance, and economics, define a sequence and its nth term, and classify finite and infinite sequences with examples like even numbers, doubling, and squares.

Learn the general term of a sequence as a formula that generates all terms, with examples a_n = 2n, a_n = 2^n, and a_n = a_{n-1}(n+3).

Explore the types of sequences, including arithmetic, geometric, and harmonic, with definitions of common difference and common ratio, and learn practical applications in finance and growth/decay.

This lecture explains arithmetic sequences, or arithmetic progressions, defined by a constant difference d, and derives the nth term formula a_n = a_1 + (n-1)d with practical examples.

Solve arithmetic sequence problems by applying the formula a_n = a_1 + (n-1)d, identifying a_1 and d, and computing terms such as the 50th, 16th, or 59th.

Apply the arithmetic sequence formula a_n = a_1 + (n-1)d to find the first term, common difference, and terms such as a_18 and a_12, with savings and deposits examples.

Explore the concept of a series as the sum of sequence terms, distinguishing finite and infinite series, with examples, and apply it to compound interest, annuities, and loan repayment formulas.

Derive the sum of n terms of an arithmetic series using a1 and d, and apply the formula Sn = n/2 (2a1 + (n-1)d).

Apply the arithmetic series sum formula to compute sums of first n terms. Solve examples including 15-term and 100-term series, and find n for a target sum like 66.

Apply arithmetic series concepts and sum formula to problems like bike depreciation over ten years, first installment in a decreasing plan, and total profit over 60 months rising by 5,000.

Explore geometric sequences (geometric progression), define the common ratio r, and derive the nth term formula a_n = a_1 r^{n-1}, with examples.

Apply the geometric sequence formula a_n = a_1 r^{n-1} to solve examples, including finding a_5 for 3, 6, 12 and first five terms for a_1 = 5, r = 2.

Apply the geometric sequence formula a_n = a_1 r^{n-1} to calculate future values. Use examples of monthly profits and town population growth to illustrate a_12 and a_7.

Derives the sum formula for a geometric series with first term a1 and common ratio r, and presents formulas for S_n across cases |r|<1, |r|>1, and r=1.

Apply the geometric series formula to two examples: a ten-term sequence with r = 2 (sum 5115) and a decreasing series with r = 1/2 (sum 1023/512).

Apply the geometric series formula to real-world problems: compute annual cumulative income from a 100 starting amount with 20% monthly growth, and total deposits doubling each year.

Learn the infinite geometric series, derive s infinity = a1/(1-r) for r<1, and note the sum exists for r<1 but not for r>1.

Apply the infinite geometric series formula to find sums, using examples like 1 + 1/3 + 1/9 + ... and 0.23 + 0.0023 + ...

Applied business mathematics introduces simple and compound interests, explains how interest rates affect the value of money, borrowing conditions, and investment scenarios, including population growth rate.

Explore fundamental terms in simple and compound interest, including principal, interest, amount, and rate, with practical examples illustrating how to calculate interest and final amount.

Explore simple interest by calculating interest on the principal and using the amount formula, with a $1,000 principal at 10% for three years yielding 1,100, 1,200, and 1,300.

Derive the simple interest formula by multiplying the principal by the rate; for n periods, simple interest equals P i N and the amount is P plus P i N.

Explore simple interest concepts through worked examples. Derive the simple interest and amount formulas from principal, rate, and time, and solve cases like loans and doubling a sum.

Explore how interest is reinvested in compound interest to grow the principal and compute the compound amount using a $1000, 10% annual rate over three years.

Derive the compound interest formula from principal, rate, and periods to show the amount as P(1+i)^n and compound interest as P[(1+i)^n-1], including semiannual, quarterly, and monthly adjustments.

Explore compound interest with a step-by-step example: $8,000 at 8% annual rate compounded semiannually over two years, showing principal, interest, and amount calculations.

Apply the compound interest formula to solve practical examples, including annual and semiannual compounding, and varying yearly rates to compute amounts and interest on principal.

Solve compound interest problems with semiannual and annual compounding, compare two investments, and analyze doubling time under simple and compound interest. Explore population growth using the compound interest formula.

Define annuities as a series of equal payments at equal intervals and study their future value and present value, including payment interval and term of the entity.

In applied business mathematics, the lecture explains two timing-based annuities—ordinary (end of period) and annuity due (beginning of period)—and three installment types: annuity certain, annuity contingent, and perpetual.

Learn to calculate the future value of an ordinary annuity by summing end-of-period payments, illustrated by investing $1,000 yearly for five years at 10%.

Derive the sum of an ordinary annuity by aggregating each year's payment grown by interest, forming a geometric series. Apply the geometric-series formula to relate r, i, and n.

Explore the sum of ordinary annuity through worked examples, calculating future values for end-of-year deposits, quarterly compounding, and monthly payments using the standard annuity formula.

Explore solving ordinary annuity problems to determine the minimum years for a $1,000 annual investment at 5% interest and the future value of a $3,000 semiannual investment at 8%.

Learn the amount or sum of annuity due, where payments at the beginning accumulate to the future value. A $1,000 yearly investment for five years at 10% illustrates the calculation.

Derive the sum of an annuity due by investing R at the beginning of each year, growing at rate I, and applying a geometric-series formula.

Explore the sum of annuity due formula to compute future values of periodic deposits. Solve monthly and quarterly examples with compounding, yielding values such as 64,500, 2,736, and 125,000.

Apply the sum of annuity due formula to solve two examples: 10,000-dollar annual payments totaling 131,800 at 6%, and a 100-per-year annuity for 15 years at 10% semi-annual.

Calculate the present value of an amount using the formula p = a / (1 + i)^n, given the rate and number of periods, and apply it to annuities.

Calculate the present value of an ordinary annuity by summing the present values of each end-of-period payment, yielding p = R(1 - (1 + i)^{-n}) / i.

Apply the present value of an ordinary annuity formula to compute present values: yearly payments of $4,000 for 10 years at 4%, and $1,000 for 3 years at 8% (semiannual).

Apply the present value of an ordinary annuity to two examples: computing a laptop's cash price from monthly payments and estimating loan payoff years.

Derives the present value of an annuity due, where payments occur at the start of each period, using a geometric series with the interest rate i.

Compute the present value of an annuity due using the formula, with start of each year payments and monthly installments for a 500,000 house at 15% annually, compounded monthly.

Explore annuity perpetual or perpetuity, its distinguishing features from other annuities, and derive the present value using P = R / i, with examples like pension and rental income.

Apply the present value of perpetuity formula to examples, converting rates to semiannual and quarterly periods, and compute P = R / i to obtain $4,000 and $2,000.