5. Formulas Flashcards
What is a Sequence?
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)
What is the Nth Term?
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.
What is the Value of a Sequence Term?
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.
What is the Sequence Rule?
- 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
What is the Direct Sequence Formula? e.g. {1, 4, 7, 10, …}
- 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
What is a Linear (or Arithmetic) Sequence?
- 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.
What is a Direct Linear (or Arithmetic) Sequence?
- 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.
If each number in a sequence is three more than the previous number, and the sixth number is 32, what is the 100th number?
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
If Sn = 3^n, what is the units digit of S(65)?
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.
What is the Recursive Sequence Formula?
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)
What is a Recursive Linear (or Arithmetic) Sequence?
- 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
What is an Exponential (or Geometric) Sequence?
A sequence in which the ratio between successive terms is always the same; a constant number (which could be negative!) is multiplied each time.
What is a Direct Exponential (or Geometric) Sequence?
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.
What is a Recursive Exponential (or Geometric) Sequence?
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.
What is a Function?
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.
