Is the parallelogram a RHOMBUS, RECTANGLE or NEITHER?

Time value of money calculations can be solved using a mathematical equation, a financial calculator, or a spreadsheet. Which of

MA_775_DIABLO [31]

To answer this question, we can assume some different possibilities for the answer, since it is incomplete (or with not clear options):

a. $\\ \frac{PMT}{r}$

b. $\\ PMT*\frac{(1+r)^{n}-1}{r}*(1 + r)$

c. $\\ PMT*\frac{(1+r)^{n} - 1}{r}$

Answer:

a. $\\ PV_{perpetuity}=\frac{PMT}{r}$

Step-by-step explanation:

The present value of a perpetuity is an amount of money needed to invest today to have a perpetuity, or an annuity paid for life, considering an interest rate of r.

PMT is a finance term for payment and r is the interest rate (roughly, an important quantity that defines how much it can be obtained for an investment).

In general, the present value can be mathematically defined as:

$\\ PV(r) = \frac{PMT_{0}}{(1+r)^{0}} + \frac{PMT_{1}}{(1+r)^{1}} + \frac{PMT_{2}}{(1+r)^{2}}+\dotsc+\frac{PMT_{n}}{(1+r)^{n}}$

Where n represents the number of periods for the investment.

On the other hand, an annuity, given a present value PV, is defined by:

$\\ PMT= A = PV*(1+r)^{n}*(\frac{r}{(1+r)^{n}-1})$

Solving this equation for PV (present value) to define the present value of an annuity, we have:

$\\ PV = \frac{(1+r)^{n}-1}{(r*(1+r)^{n})}*PMT$

But the question is asking for an annuity paid for life (theoretically, for infinite periods of time); then, if we calculate the limit for the previous equation when n tends to infinity, we find that:

$\\ lim_{n\to\infty} \frac{(1+r)^{n}-1}{(r*(1+r)^{n})}*PMT$

$\\ (lim_{n\to\infty} \frac{(1+r)^{n}}{r*(1+r)^{n}} - lim_{n\to\infty} \frac{1}{r*(1+r)^{n}})*PMT$

$\\ (lim_{n\to\infty} \frac{(1+r)^{n}}{(1+r)^{n}}*\frac{1}{r} - lim_{n\to\infty} \frac{1}{r*(1+r)^{n}})*PMT$

$\\ (lim_{n\to\infty} 1*\frac{1}{r} - lim_{n\to\infty} \frac{1}{r*(1+r)^{n}})*PMT$

The second term of the previous expression tends to 0 (zero) when n tends to infinity, then:

$\\ (lim_{n\to\infty} 1*\frac{1}{r})*PMT$

$\\ (1*\frac{1}{r})*PMT$

$\\ \frac{PMT}{r}$ or

$\\ PV_{perpetuity}=\frac{PMT}{r}$

This expression represents that, with an interest of r, if we make an investment of PMT today, then we will have an annuity of $\\ \frac{PMT}{r}$ for life, because in each period PMT would be the same again due to the interest rate (r).

6 0

3 years ago

HELP MEEE please NO BOTS

neonofarm [45]

Answer:

64

Step-by-step explanation:

You'd put 64 in the box because the x is the original number (4). example: 4^3 = 64, ^3(insert the root sign) 64 = 4

sorry in advanced, I dont have the math signs to make the equations

6 0

1 year ago

Write a recursive formula for the following arithmetic sequence. 8, 5, 2, −1,