Use the present value formula to determine the amount to be invested now, or the present value needed. The desired accumulated amount is $140,000 after 2 years invested in an account with 8% interest compounded quarterly.

The amount to be invested now, or the present value needed, is $
(Round to the nearest cent as needed.)

The value needed now is $102,296.60

Step-by-step explanation:

Here, we want to calculate the present value needed.

We shall be using the formula for compound interest here

Mathematically;

A = I(1 + r/n)^nt

According to the question;

I is the initial amount which is what we want to calculate

A is the accumulated amount with interest = 140,000

r is the interest rate = 8% = 8/100 = 0.08

n is the number of times interest is compounded which is 4, since it is quarterly

t is the number of years = 2

Substituting these values, we have ;

140,000 = I(1 + 0.08/2)^(4 * 2)

140,000 = I(1.04)^8

I = 140,000/(1.04)^8

I = 102,296.62870027774

Which is I = $102,296.60 to the nearest cent

geometric series converge the common ratio r has magnitude less than one.

which is equivalent to

for the example we need

|2-x| < 1

-1 < 2-x   and 2-x < 1

x < 3 and x > 1

we can write that

answer: 1 < x < 3

1a. common ratio r=250/25=10 > 1     diverges

1b.   r=50/100=1/2 converges

when these converge they converge to

where a is the first term.

answer: 200

1c. r=-1   so it's not the case that |r| < 1.     diverges

2a. x is the common ratio so we converge when |x|< 1

2b.   r=-10x / -2 = 5x.

we converge when

|5x| < 1

|x| < 1/5

2c.   r=3-2x

-1 < 3-2x and 3-2x < 1

-4 < -2x and 2 < 2x

x < 2 and 1 < x

1 < x < 2

2d. [/tex] r=-48x^3/12 = 4x^3[/tex]

|4x^3| < 1

|x^3| < \frac 1 4

|x| < 4^{-1/3}