What is the sum of the first five terms of the geometric sequence in which a1 = 10 and r = 1/2?

Question

7. What is the sum of the first five terms of the geometric sequence in which a1 = 10 and r = 1/2?


8. The original purchase price of a car is $12,000. Each year, its value depreciates by 5%. Three years after its purchase, what is the value of the car?
A. $1.50
B. $13,891.50
C. $10,288.50
D. $10,830

9. Which of the following sequences are geometric? Check all that apply.
A. 2, 5, 8, 11, 14, 17
B. 2, 3, 5, 8, 13, 21
C. -4, -2, -1, -0.5, -0.25, -0.125
D. 6, 18, 54, 162, 486

10. Which of the following are true statements? Check all that apply.
A. logM-logN=log(M-N)
B. logM+logN=log(MN)
C. logM^p=p*logM
D. (logM)^p=p*logM

11. Simplify the expression below and write it as a single logarithm: 3log(x+4)-2log(x-7)+5log(x-2)-log(x^2)
A. Log((x+4)^3(x-2)^5/(x-7)^2x^2)
B. Log(x^2(x+4)^3(x-7)^5/(x-2)^5)
C. Log((x+4)^3(x-7)^5/(x-2)^5x^2)
D. Log((x+4)^3(x-7)^2/(x-2)^-5x^-2)

Answer 1

7. Using Brute Force, the first five terms are 10, 5, 2.5, 1.25, and 0.625. Add these together and you get 19.375. That's not an elegant way to come up with an answer, though. We can apply the formula:

S=a1(1-rⁿ)/(1-r) = 10 * (1-(0.5)⁵)/(1-0.5) = 10 * (1-0.03125)/0.5 = 19.375!

 

8. You can rule out (A). That's just too small of a value for three years of depreciation . You can also rule out (B). It's depreciation, not an investment! So we need to figure out whether its (C) or (D).

 

After each year, the car loses 5% of its value. An equivalent expression is "at the end of each year, the car is worth 95% of what it was worth at the start of that year."

After the first year, the $12000 car is worth 0.95*12000 = 11400.

After the second year the car is worth 0.95*11400=10830.
After the third year, the car is worth 0.95*10830=10288.5.  So the answer is (C)

 

That method was also clunky.  What if we used a formula that captured the repetitive calculation we just did as an exponent? Like: Present Value=Initial Value (1*rate)^n, where "rate" is the annual depreciation rate, and "n" is the number of years. Present Value=$12000 (1-0.05)³ = $12000 (0.95)³ =$12000 * .857375 = $10288.50!

 

9. A geometric sequence is defined as a sequence of numbers in which each term after the first is found by multiplying the previous by a fixed constant. We need to examine each sequence for these properties.

A is actually an arithmetic sequence, found by ADDING 3 to get successive terms.

B is not a geometric sequence. 3 and 5 - the 2nd and 3rd terms - are prime numbers. This is actually part of the Fibbonacci sequence!

C is indeed a geometric sequence. Each successive term after the first can be found by multiplying the previous one by 1/2. 

D is also a geometric sequence. Here the common factor is 3.

10. For this question, you need to know the basic rules of arithmetic for logarithms.

A is FALSE because logM-logN=log(M/N)

B is TRUE

C is TRUE

D is FALSE (logM)^p cannot be simplified. Logarithms, like any other function, can be raised to powers!

 

11. The answer is A. The positive coefficients go in the numerator of the log and the negative coefficients go in the denominator. Coefficients of the logarithm become powers of its argument.