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1
Q

What is a Sequence?

A

A collection of numbers in a set order. {1, 4, 7, 10, …} is an example of a sequence for which the first four terms are specified (but the sequence continues beyond these four terms, as indicated by the “…”)

Every sequence is defined by a rule (e.g. An = 9n + 3), which you can use to find the values of terms. You can find the nth term (An) by plugging n into the equation)

2
Q

What is the Nth Term?

A

A particular term in a sequence. The term number, n, corresponds to the term’s location in a sequence. In the sequence {1, 4, 7, 10, …}, 1 is the 1st term (n = 1), 4 is the 2nd term (n = 2), 7 is the 3rd term (n = 3), and so on.

3
Q

What is the Value of a Sequence Term?

A

The value of a particular term in the sequence. In the sequence {1, 4, 7, 10, …} the 1st term has a value of 1, the 2nd term has a value of 4, the 3rd term has a value of 7, and so on.

4
Q

What is the Sequence Rule?

A
  • The rules that determine the order of numbers in a given sequence. In the sequence {1, 4, 7, 10, …}, each term is 3 more than the previous term, so the rule is to add 3 each time to get the next term
  • Rules can also be written as direct or recursive sequence formulas
5
Q

What is the Direct Sequence Formula? e.g. {1, 4, 7, 10, …}

A
  • One way to write a sequence formula
  • A direct sequence is defined as a function of n, the place in which the term occurs in the sequence
  • For the sequence {1, 4, 7, 10…}, the direct sequence formula is An = 3n – 2, for integers n ≥ 1
6
Q

What is a Linear (or Arithmetic) Sequence?

A
  • A sequence in which the difference between successive terms is always the same
  • A constant number (which could be negative!) is added each time
  • Also called Arithmetic Sequence.
7
Q

What is a Direct Linear (or Arithmetic) Sequence?

A
  • One way to write the direct linear sequence formula is Sn = kn + x where k is the constant difference between successive terms, x is some other constant, and n is the number of the term in question
  • Another way to write the direct linear sequence formula is Sn = S1 + (n - 1)k, where S1 is the value of the first term in the sequence, n is the number of the term in question, and k is the constant difference between successive terms.
8
Q

If each number in a sequence is three more than the previous number, and the sixth number is 32, what is the 100th number?

A

Instead of finding the rule for this sequence, consider the following reasoning:

From the sixth to the one hundredth term, there are 94 “jumps” of the 3. Since 94 * 3 = 282, there is an increase of 282 from the sixth term to the one hundredth term:

32 + 282 = 314

9
Q

If Sn = 3^n, what is the units digit of S(65)?

A
Clearly, you cannot be expected to multiply out 3^65 on the GMAT. Therefore, you must assume that there is a pattern in the powers of three. 
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243
3^6 = 729

Note the pattern of the units digits in the powers of 3: 3, 9, 7, 1 [repeating]…Also note that the units digit of Sn when n is a multiple of 4 is always equal to 1. You can use the multiples of 4 as “anchor points” in the pattern. Since 65 is 1 more than 64 (the closest multiple of 4), the units digit of S(65) will be 3, which always follows 1 in the pattern.

10
Q

What is the Recursive Sequence Formula?

A

Another way to write a sequence formula. A recursive sequence is defined in terms of the value of previous items in the sequence. For the sequence (1, 4, 7, 10, …}, the recursive sequence formula is An = An-1 + 3.

*Keep track of the terms you know on your scrap paper! (e.g. you know a6 but do not know a5 or a4)

11
Q

What is a Recursive Linear (or Arithmetic) Sequence?

A
  • The recursive linear sequence formula is Sn = Sn-1 + k, where Sn-1 is the value of the previous term in the sequence and k is the constant difference between successive terms
  • In addition to the recursive formula, the value of one specific term must be given, along with its term number. For example, S2 = 6 tells us that the 2nd term of the sequence has the value 6.

*NOTE: when a sequence is defined recursively, the question will have to give you the value of at least one of the terms; those values can be used to find the value of the desired term

12
Q

What is an Exponential (or Geometric) Sequence?

A

A sequence in which the ratio between successive terms is always the same; a constant number (which could be negative!) is multiplied each time.

13
Q

What is a Direct Exponential (or Geometric) Sequence?

A

The standard formula is Sn = x(k^n), where x is the value of the first term in the sequence, k is the value of the ratio (the number by which we multiply each successive term), and n is the number of the term in question.

14
Q

What is a Recursive Exponential (or Geometric) Sequence?

A

The standard formula is Sn = (k)(Sn-1), where k is the value of the ratio (the number by which we multiply each successive term), and n is the number of the term in question. In addition to the recursive formula, the value of one specific term must be given, along with its term number. For example, S2 = 6 tells us that the 2nd term of the sequence has the value 6.

15
Q

What is a Function?

A

A rule, or formula, which takes an input (or given starting value) and produces an output (or resulting value). For example, f(x) = x + 3 represents a function, where x is the input, f(x) is read as “f as a function of x” or “f of x” and refers to the output (also known as the “y” value), and x + 3 is the rule for what to do to the x input. f(4) = x + 3 = 4 + 3 = 7.

16
Q

How do you solve functions with unknown constants? For example, if f(x) = ax^2 – x and f(4) = 28, what is f(-2)?

A

(1) Use the value of the input variable and the corresponding output variable of the function to solve for the unknown constant
f(4) = a(4)^2 – 4 = 28
16a – 4 = 28
a = 2

(2) Rewrite the function, replacing the constant with its numerical value
f(x) = ax^2 – x = 2x^2 – x

(3) Solve for the function for the new input variable
f(-2) = 2(-2)^2 – (-2) = 10

17
Q

What is an Independent Variable?

A

The input variable of a function. In the function f(x) = x + 3, x represents the independent variable.

18
Q

What is a Dependent Variable?

A

The output variable of a function. In the function f(x) = x + 3, f(x) represents the dependent variable, while x by itself represents the independent variable. f(x) does not mean “f times x” and the letter f is not a variable; rather, it is read as “f as a function of x” or “f of x.”

19
Q

What is the Domain?

A

All of the possible inputs, or numbers that can be used for the independent variable, for a given function. In the function f(x) = x^2, the domain is all numbers.

20
Q

What is a Range?

A

All of the possible outputs, or numbers that can be used for the dependent variable, for a given function. In the function f(x) = x^2, the range is f(x) >= 0.

21
Q

What are Compound, or Composite, Functions?

A

Two nested functions; solved from the inner parentheses out. For example, f(g(x)) is an example of a compound function and is read as “f of g of x.” Given f(x) = x + 3 and g(x) = 2x, g(x) is substituted first: f(g(x)) = f(2x). Next, f(x) is substituted: f(2x) = 2x + 3.

22
Q

What is Direct Proportionality?

A
  • Two quantities always change by the same factor and in the same direction
  • For example, doubling the input causes the output to double as well
  • Standard Formula: y = kx, where x is the input, y is the output, and k is the proportionality constant (or the factor by which the numbers change)
  • Can also be written as (y/x) = k, which means that the ratio of y to x is always the same constant.
23
Q

The maximum height reached by an object thrown directly upward is directly proportional to the square of the velocity with which the object is thrown. If an object thrown upward at 16 feet per second reaches a maximum height of 4 feet, with what speed must the object be thrown upward to reach a maximum height of 9 feet?

A

Typically with direct proportion problems, you will be given “before” and “after” values. Simply set up ratios to solve the problem

H1/(V1)^2 = Before
H2/(V2)^2 = After
H1/(V1)^2 = H2/(V2)^2 since both ratios are equal to the same constant k
(4)/(16^2) = 9/(V2)^2
(V2)^2 = 9 (16^2 / 4) = 576
V2 = 24

The object must be thrown upward at 24 feet per second

24
Q

What is Inverse Proportionality?

A
  • Two quantities change by reciprocal factors
  • For example, cutting the input in half causes the output to double
  • Standard Formula: y = (k/x), where x is the input, y is the output and k is the proportionality constant
  • Can also be written as xy = k, which means that the product of y and x is always the same constant.
25
Q

The amount of electrical current that flows through a wire is inversely proportional to the resistance in that wire. If a wire currently carries 4 amperes of electrical current, but the resistance is then cut to one-third of its original value, how many amperes of electrical current will flow through the wire?

A

While you are not given precise amounts for the “before” or “after” resistance in the wire, you can pick numbers. Using 3 as the original resistance and 1 as the new resistance, you can see that the new electrical current will be 12 amperes.
C1 * R1 = C2 * R2
4(3) = C2(1)
C2 = 12

26
Q

What does Proportionality Constant mean?

A

The constant by which proportional values, whether direct or indirect, (above) change.

27
Q

What is Linear Growth or Decay?

A
  • Standard Formula: y = mx + b, where x is the input, y is the output, slope m is the constant rate at which the quantity grows or declines, and b is the value of the quantity at time zero
  • Linear growth refers to a quantity that grows at a constant rate; a positive constant is added for each given period of time
  • Linear decay refers to a quantity that shrinks at a constant rate; a negative constant is added for each given period of time
28
Q

Jake was 4.5 feet tall on his 12th birthday, when he began to have a growth spurt. Between his 12th and 15th birthdays, he grew at a constant rate. If Jake was 20% on his 15th birthday than on his 13th birthday, how many inches per year did Jake grow during his growth spurt?

A
Linear Growth formula = y = mx + b
12th Birthday (x = 0): 4.5 feet = 54 inches = b
13th Birthday (x = 1): 54 + m
14th Birthday (x = 2) 54 + 2m
15th Birthday (x = 3) : 54 + 3m

54 + 3m = 1.2(54 + m)
54 + 3m = 64.8 + 1.2
1.8m = 10.8
m = 10.8/1.8 = 6

y = 6x + 54, Jake grew 6 inches per year

29
Q

What is Exponential Growth or Decay?

A
  • In exponential growth, a quantity is multiplied by the same constant each period of time (rather than adding the same constant, as in linear growth)
  • Any exponential growth can be written as Y(t) = Y0 * k^t, in which Y represents the quantity as a function of time t, Y0 is the value of the quantity at time t = 0, and k represents the constant multiplier for one period

NOTE: The ratio of the values in any two consecutive periods is constant

30
Q

What are Maximums or Minimums?

A

The maximum possible output or minimum possible output of a given function. For example, the minimum output for the function f(x) = x^2 is 0. The maximum output is infinity.

31
Q

For which of the following functions does f(x – y) NOT EQUAL f(x) – f(y)?

A

For all GMAT problems that test whether certain functions follow certain properties of mathematics, the most effective approach is to simply pick numbers and see which function gives the desired result

32
Q

Algebra Guide, Ch 10, Q 2. If g(x) = 3x + √x, what is the value of g(d^2 + 6d + 9)?

A

= 3(d^2 + 6d + 9) + √(d + 3)^2
= 3d^2 + 18d + 27 + d + 3
= 3d^2 + 19d + 29

33
Q

Algebra Guide, Ch 10, Q 7. The velocity of a falling object in a vacuum is directly proportional to the amount of time the object has been falling. If after 5 seconds, an object is falling at a speed of 90 miles per hour, how fast will it be falling after 12 seconds?

A

V1/T1 = V2/T2
90mph/5 seconds = V2/12 seconds
V2 (12 seconds)(90 mph) / 5 seconds = 216 mph

34
Q

Algebra Guide, Ch 10, Q 9. A strain of bacteria multiplies such that the ratio of its population in any two consecutive minutes is constant. If the bacteria grows from a population of 5 million to a population of 40 million in one hour, by what factor does the population increase every 10 minutes?

A

This is an exponential growth problem, so you should use the equation Y(t) = Y0 * k^t

In 60 minutes, the population grew by a factor of 40/5 = 8, which also equals k^60

Thus, in one minute, the population grows by a factor of k = 8^(1/60) = (2^3)^(1/60) = 2^(1/20)

The question asks what factor the population will grow by in 10 minutes, so simply take this factor to the tenth power. k^10 = (2^(1/20))^10 = √2

35
Q
Algebra Guide, Ch 10, Q 10. For which of the following functions does f(x) = f(2 – x)?
A. f(x) = x + 2
B. f(x) = 2x – x^2
C. f(x) = 2 – x 
D. f(x) = (2 – x)^2
E. f(x) = x^2
A

This is a symmetry function type of problem. Generally, the easiest way to solve these kinds of problems is to pick numbers and plug them into each function to determine which answer gives you the desired result. For example, pick x = 4
A. f(4) = 4 + 2 = 6 VS f(2 – 4) = (2 – 4) + 2 = 0 — NO
B. f(4) = 2(4) – (4)^2 = -8 VS f(2 – 4) = 2(2 – 4) – (2 – 4)^2 = -8 — CORRECT
C. f(4) = 2 – 4 = -2 VS f(2 – 4) = 2 – (2- 4) = 4 — NO
D. f(4) = (2 – 4)^2 = 4 VS f(2 – 4) = [2 – (2 – 4)]^2 = 16 — NO
E. f(4) = 4^2 = 16 VS f(2 – 4) = (2 – 4)^2 = 4 — NO

36
Q

What are common patterns seen in GMAT sequences?

A

(1) Repeats: 1, 3, -2, 1, 3, -2, etc
(2) Consecutive Integers: 10, 11, 12, 13, etc
(3) Consecutive Multiples: 7, 14, 21, 28, etc
(4) Evenly Spaced Sets: 9, 16, 23, 30, etc
(5) Non-Uniform Spacing: 0, 1, 3, 6, 10, 15, etc
(6) Alternating Signs: -1, 1, -2, 2, etc