Sequences and Formulas Flashcards
Plug-In Formulas:
The most basic GMAT formula problem. You get a formula with multiple variables, get some values and then have to solve for the unknown variable in the formula. Write the formula down as a whole equation on your scrap paper, not just the formula.
E.g. C = QL/J
Formulas problems are only tricky because the given formulas are unfamiliar. But the calculation is usually straight-forward once you have filled in all the values.
Functions:
Functions in the form of this:
f(x) = 2x + 3
are rules built on an independent variable, here x. The value of the function, here f, changes as the value of the variable changes because the value of the function is dependent on the value of the variable.
Another example for function:
g(t) = t^3 + √t - 2t/5 the value of function g is dependent on independent value t.
You can think of a function as a box that you put a variable into and it turns the variable into some output value.
The DOMAIN of a function indicates the possible inputs. The RANGE indicates the possible outputs. E.g. function f(x)= x^2 can take any input but never produces a negative number, so domain is all numbers, range is f(x)≥0.
Usually you get a value for the variable and have to calculate the output. Sometimes GMAT will not give you a numerical input, but an input that’s an algebraic expression.
E.g. If f(z) = z^2 - z/3, what is the value of f(3w+6)?
Just put the value (3w+6) into the function and solve. Be aware of negative signs and how they affect expressions.
Strange Symbol Formulas:
GMAT can use strange symbol and use it to define a certain procedure. It is helpful to break the operation down one by one and understand which variable stands for which value.
E.g. for formulas:
x # y = x^2 + y^2 -xy
or
W @ F = (√F)^w for all integers W and F. What is 4 @ 9?
Pay attention to where the variables are, left-right etc., and use the right values in right spots.
W @ F = (√9)^4 = 3^4 = 81
Some questions may require you to use the given procedure more than once.
E.g. W @ F = (√F)^w for all integers W and F. What is 2 @ (3 @ 16)?
Perform the procedure inside parentheses first:
W @ F = (√16)^3 = 4^3 = 64
Then do the second procedure:
W @ F = (√64)^2 = 8^2 = 64
Formulas That Act on Decimals:
In these follow instructions PRECISELY!
Symbol [x] represents the largest integer less than or equal to x. That means here you have a value inside the square brackets for the variable x. In this case it will be a decimal number. The function of the brackets puts out a number that’s an integer less than or equal to that value you put in for x.
E.g. What is [5.1]
x is 5.1 and the function is defined as a value that is an integer less than or equal to that number. So the output here would be 5.
Be careful with negative decimals.
E.g. What is [-2.3]
Output would be -3 because that is the largest integer less than or equal to -2.3. It wouldn’t be -2 because the function says “less than.”
Sequence Formulas:
A sequence is defined by the rule you are given in the formula.
E.g. Sn = 15n - 7. What is value of S7 - S5?
S7 = 15x7 - 7 = 98 S5 = 15x5 - 7 = 68
S7 - S5 = 98-68 = 30
Sometimes the sequence will be defined RECURSIVELY. That means each term is defined relative to other terms. You will be given one value and have to find another one.
E.g. If An = 2An-1 - 4, and A6 = -5, what is value of A4?
Pay attention to what the definition really tells you. Here the An represents the nth term, so An-1 represents the term right before it. You are given the 6th term, A6, and are looking for the 4th term, A4. But first you need to find A5. Write done what you have in this stile to keep track:
—— = A4 —— = A5 -4 = A6
Fill in the value of A6 to find the value for A5. A6 is An and we want to find the value right before A6, namely A5.
A6 = 2An-1 - 4 -4 = 2 A6-1 - 4 -4 = 2 A5 - 4 0 = 2 A5 0 = A5
Now you know A5 is 0. You can find A4 now by repeating the process.
A5 = 2A4 - 4 0 = 2A4 - 4 4 = 2A4 2 = A4
In Linear Sequence problems you may be given a word problem that’s a formula/sequence problem but instead of finding the rule you could just look the issue described and find a reasonable way to get to solution.
E.g. If each number in a sequence is 3 more than the previous, and the 6th number is 32, what is the 100th number?
You know that each number is increased by 3 and that you have 94 numbers to get to 100. So the calculation is:
32 + 94 x 3 = 314
Functions and Symbolism on GMAT:
GMAT uses classic function notations like these:
f(x) = X^2 - 1.
This basically means that whatever goes in between the parentheses gets substituted in place of x in x^2.
GMAT also uses strange symbols as substitutions for either operations (most of the time) or rarely also digits/numbers.
E.g. if x @ y = 3x - y^2, then what is the value of 8 @ 2.
Whatever is on left of the @ symbol is x and should be substituted for x in 3x - y^2 and whatever is on the right is of @ is y and should be substituted for y.
x @ y = 3x - y^2
8 @ 2 = 3*8 - 2^2
8 @ 2 = 24 - 4
8 @ 2 = 20
In symbolism questions directly apply the definition of the symbol to the numbers provided. The definition gives you an equation. Then you are given an if equation of which you must take the values provided and put them into the definition, so you must substitute the values given in the “if equation” into the defining equation. You must also set the defining equation equal to the value given in the “if equation” and then solve it for the variable you are asked to solve.
Functions and Symbolism on GMAT - Example for Operations:
Symbolism questions are usually solved by substitution.
Example for when symbol is substituted for operation:
The Symbol # represents one of the following operations: addition, subtraction, multiplication, or division. What is the value of 6 # 2?
(1) 0 # 3 = 0
(2) 2 # 1 = 2
Your task is to find out what operation the symbol # represents so you can solve the question declare sufficiency.
Statement 1:
The symbol could be multiplication or division so it’s not sufficient.
Statement 2:
The symbol could be multiplication or division so not sufficient.
Statements together: If taken together you still only know that the symbol could either be multiplication or division so the answer here would be E.
Nested Functions in Symbolism Problems
In symbolism problems you are sometimes asked to solve nested functions, where one function is put into another and then that one into another etc. With nested functions you have to start with the innermost function and then work your way out.
E.g. if you are given equations and then told to solve this:
Variable m inside a circle inside a square (can’t draw it here) and you are given the value for m then you have to plug m into the circle equation and solve for the value and then put that value into the square equation and solve for the final value.
Sequence and Series Questions:
In sequence and series questions there is usually a pattern. Your job is to find the pattern by testing numbers and then finding the solution based on that pattern.
E.g. Sn = -1/Sn-1 + 1 for all integer values of n greater than 1. If S1 = 1, what is the sum of the first 61 terms in the sequence?
You don’t have time to find the first 61 terms and add them up. So you will have to find a pattern in the sequence and thereby get to the result.
S1 = 1 is given
Now the best way to solve this issue is to calculate a few values of S by plugging in consecutive numbers for n to look for the pattern:
S2 = -1/S2-1 + 1 = -1/S1 + 1 = -1/1 + 1 = -1/2
S3: -1/S2 + 1 = -1/-1/2 + 1 = -1/1/2 = -1 x 2 = -2
S4: -1/S3 + 1 = -1/-2 + 1 = -1/-1 = 1
S5: -1/S4 + 1 = -1/1 + 1 = -1/2
S6: -1/S5 + 1 = -1/-1/2 + 1 = -1/1/2 = -1 x 2 = -2
By now you can see that the pattern for n is: 1, -1/2, -2, 1, -1/2, -2 etc.
That means there are 3 numbers in a group that repeats itself over and over. Now we have to find the sum for the first 61 terms:
The sum for the first three terms: 1 + (-1/2) + (-2) = -3/2
for the fist 60 terms you can calculate: (-3/2) x 20 = -30
for the 61st term add a 1 because that’s the result for the first in the group of 3: -30 + 1 = -29
NOTE: If you don’t spot a pattern within the first 5 - 8 terms in sequence and series questions stop using this approach and see if there’s another way (including guessing).
Extra Formulas Strategy: Look for Patterns in Sequence Questions:
Some sequences are easier solved by looking for patterns.
E.g. if Sn = 3^n what is units digit of S65?
Look at powers of 3 to find the pattern:
3^1 = 3 3^2 = 9 3^3 = 27 3^4 = 81 3^5 = 243 3^6 = 729
So the pattern for the units digit of powers of 3 is: 3, 9, 7, 1, [repeating]
Since this is a group of 4 you can divide 65 by 4 and see what the remainder is. Result is 64 and remainder of 1 which means the units digit of S65 will be 3.
REMEMBER for Sequences on GMAT:
Sequences are defined for n ≥ 1, meaning it will always start with S1. If not they will tell you it’s defined for n ≥ 0 in which case you start with S0. The second term will be S1.
Extra Formulas Strategy: Compound Functions:
Compound functions give you two rules to use. Key is to work from inside out.
E.g. if f(x) = x^3 + √x and g(x) = 4x - 3 what is f(g(3))?
Expression f(g(3)) means “f of g of 3”
g (3) = 4 x 3 - 3 = 9
f (9) = 9^3 + √9 = 729 + 3 = 732
In compound functions pay attention to the order you perform the calculations: from inside out!!
GMAT may ask you to find the value of x for certain given functions. First find the expressions as given by the function rules and then solve for x:
E.g. if f(x) = x^2 + 1, and g(x) = 2x, for what positive value of x does f(g(x)) = g(f(x))?
f(g(x)) = (2x)^2 + 1 = 4x^2 + 1 g(f(x)) = 2 (x^2 + 1) = 2x^2 + 1
Not set equal to solve for x:
4x^2 + 1 = 2x^2 + 1 2x^2 = 1 x^2 = 1/2 x = √1/2 take only positive root because x must be positive according to stem.
Functions With Unknown Constants:
GAMT may give you a function with an unknown constant. You will also be given the value of the function for a specific number.
E.g. if f(x) = ax^2 - x and f(4) = 28, what is f(-2)?
First put 4 and 28 into function and solve for a. Once you know a you can then solve for f(-2).
Population Problems:
In these create a population chart and do the computation.
E.g. the population of a certain type of bacterium triples every 10 minutes. If the population was 100 20 minutes ago, how many minutes from now approximately will it be 24,000?
Create population chart to show numbers 20 mins ago, 10 mins ago, now, 10 mins from now…etc. Answer is approximately 30 mins from now.
In some cases you will have to pick a number as the starting point in these questions. Pick smart numbers that make computation easy.
Proportionality in Function Questions:
In many GMAT questions with functions you’ll have direct or inverse proportionality.
- Direct proportionality means that for instance when you triple the input (x), your output (y or f(x)) will triple as well. The relationship here is:
y = kx where k is a constant. That also means: y/x = k which means that the ratio of the output and in input always yields the same constant.
E.g. 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 an object be thrown to reach maximum height of 9 feet?
Not the that height is proportional to the square of the velocity, not the velocity itself.
You can set up ratios to solve:
h1/(v1)^2 = h2/(v2)^2
4/(16)^2 = 9/(v2)^2 4 = 9 x (16)^2/(v2)^2 4 = 9 x 256/(v2)^2 4x(v2)^2 = 9 x 256 (v2)^2 = 9 x 256/4 (v2)^2 = 9 x 64 v2 = √9 x 64 v2 = 3 x 8 v2 = 24
- Inverse proportionality means that when you cut the input in half, your input doubles, if you triple the input, the output is cut to one-third of its value. So, the quantities change by reciprocal factors.
That means: xy = k
In these problems you set up product, not ratios, to solve. Because you have x1y1 = k and x2y2 = k you can set equal:
x1y21 = x2y2
E.g. the amount of electrical current that flows through a wire is inversely proportional to the resistance in the 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?
You can pick number for the before and after resistance. So, let’s say for before resistance 3 and after resistance 1.
Therefore:
c1r1 = c2r2 4x3 = c2x1 12 = c2
- Linear Growth means growth at a constant rate. The growth is determined by the linear function y = mx + b where m, the slope, is the constant rate at which the quantity grows, and b is the quantity at time 0, that means when x is 0, i.e. when no growth has yet taken place. E.g. if a baby is born at 9 pounds and gains 1.2 pounds per month, the baby’s weight is expressed as: W = 1.2 x + 9, where x is the amount of months that have passed.
E.g. Jake was 4.5 feet tall on this 12th bday when he began to have a growth spurt. Between his 12th and 15th bday he grew at a constant rate. If Jake was 20% taller on his 15th bday than on his 13th bday, how many inches per year did he grow during his growth spurt? (12 inches = 1 feet)
First convert all numbers into inches as that’s what the answer should be.
- 5 feet = 4.5 x 12 inches
- 5 feet = 54 inches.
Jake is 54 inches tall on his 12th bday, i.e. at time 0, so:
mx0 + 54 = 54 inches (Jake’s 12th bday)
On his 13th bday x = 1, on his 14th bday x = 2, and on his 15th bday x = 3.
What we are looking for is m, the growth rate.
If on his 15th bday he is 20% taller than on his 13th bday we can write:
mx3 + 54 = (mx1 + 54) + 0.2 (mx1 + 54) mx3 + 54 = 1.2 (mx1 + 54) mx3 + 54 = 1.2m + 64.8 1.8m = 10.8 m = 10.8/1.8 m = 108/18 m = 6
So Jake grew 6 inches per year.
- Symmetry means that two different inputs yield the same output. In these picking a number and doing the calculation to see if the two values yield the same output is the easiest.
E.g. For which of the following functions does f(x) = f(1/x) if x≠-2, -1, 0, or 1.
E.g. you can pick 2 and for each of the five functions given in the answer choices calculate the value for f(2) and f(1/2) and see for which function the result is the same.
The result will show that f(x) = f(1/x) for function f(x) = |x+1/x-1|
Here it is important to remember that signs in absolute value brackets don’t matter so if you have f(x) = |2| and f(1/2) = |-2| it is the same result, 2 because in absolute value questions the distance from 0 is the result.
You can also solve the question algebraically by substituting 1/x into each of the answer choice functions and seeing where you get the same function as was given in the answer choice.
E.g. if you put 1/x into f(x) = |x+1/x-1|
|1/x + 1/1/x -1| = |1/x + x/x/ 1/x - x/x| = |1+x/x/1-x/x| = |(1+x/x)(x/1-x)| = |1+x/1-x| = |1+x/-1(1-x)| = |1+x/-1 +x| = |x+1/x-1|
Here it is important to remember that because of the absolute value brackets it is permitted to multiply by (-1) to manipulate the expression to get what you want.
