10th Grade Math Reviewer PDF

1st Quarter Math Reviewer By: Trinidad Villongco FORMULAS Arithmetic Sequence Sequence A set of numbers arranged by some fixed rule. Arithmetic Sequence A list of numbers in which two consecutive terms have a constant difference. Common Difference...

1st Quarter Math Reviewer By: Trinidad Villongco FORMULAS Arithmetic Sequence Sequence A set of numbers arranged by some fixed rule. Arithmetic Sequence A list of numbers in which two consecutive terms have a constant difference. Common Difference The fixed number is added or subtracted from the preceding term to get the next term in an arithmetic sequence. Fixed difference between two consecutive terms. Example: 1. Write the first 5 terms for the following sequence. 2. Find the common difference. Formula for finding the nth term Examples 1. Find the 20th term of an arithmetic sequence given the first term – 5 and the common difference – 3. 2. Find the 13th term of the following sequence: 7, 17, 27 … 3. Which term in the arithmetic progression 4,1, - 2,... is – 77? We are looking for the number of terms. Currently, we have the nth term, -77, the first term, 4, and the common difference, -3, which we can get by subtracting 4 from 1. All you have to do is substitute the terms in the formula with our numbers then boom bam. In the second line, I distributed the (n-1) to -3 so it's easy to solve. 4. The 3rd term of a sequence is 94 and the 6th term is 85. Find the 26th term. Since we don’t have the first term, number of terms, or common difference we can start solving this problem by getting the common difference. I got the common difference by using our a = a₁ + (n-1)d formula but substituting the nth term for the 6th term and the first term for the 3rd term. To get the number of terms, I subtracted 3 from 6 to get the exact number of terms in our equation because here 3 is technically our first term and 6 is our last term. From there you can just solve it. Once you have a common difference, solve for the 26th term by substituting the last term for the 26th term and the first term for either the 3rd or 6th term, whichever you prefer. I did the same thing for getting the number of terms in looking for the 26th term as I did while looking for the common difference. Arithmetic Means Arithmetic Means If a, A, and b are three consecutive terms in an arithmetic sequence, then A is called the arithmetic mean of a and b. Examples 1. Find the arithmetic mean of 17 and – 3 2. Insert two arithmetic means between −5 and 40. To insert more than one arithmetic mean, you should look for the common difference between the two numbers. 3. Find the value of x when the arithmetic mean of x + 2 and 4x + 5 is 3x + 2. Since it gave us the arithmetic mean, a, and b, I just substituted. After that, I added/subtracted all the like terms. Then, I multiplied both sides by 2 to cancel out the 2 on the right side and distributed the 2 on  the left side. I transmuted 5x to the left side and 4 to the right then just solved. 4. Insert two arithmetic means between x + y and x − 2y Arithmetic Series Arithmetic Series The partial sum of an arithmetic sequence Formula for Arithmetic Series Examples 1. Find the sum of the following 20 terms 60 + 64 + 68 + 72 … 2. Find the sum of the arithmetic series. 100 + 105 + 110 + … 220 3. The Seventh term and the twelfth term are 28 and 73 respectively. a. Find the common difference.  b. Find the first term. c. Find the sum of the first 40 terms. Sigma Notation (Arithmetic) A simple way of indicating the sum of a finite number of terms in a sequence is the summation notation. Adding up the terms of a sequence. Examples 1. 2. Geometric Sequence Geometric Sequence A type of number sequence in which the ratio of two consecutive terms is always constant. Common Ratio Ratio of two consecutive geometric terms The fixed number being multiplied/divided to the preceding term Geometric Sequence Formula Examples 1. Find the indicated term of a geometric sequence: a20: a1 = 3, r = 2  2. Find the 12th term of a geometric sequence if the first term is 2 and the third term is 50. Geometric Mean Geometric Mean Average that multiplies all values and finds the root of the product. Formula for Geometric Mean a and b are consecutive terms of a geometric sequence Examples 1. Find the geometric mean of 9 and 81 2. Insert two geometric mean for the geometric sequence 7,____,____, 189. Geometric Series Geometric Series The sum of the finite/infinite terms of a geometric sequence. Formula for Geometric Series Examples 1. Determine the sum of the first 10 terms of the geometric series: 4 + 12 + 36 + …  2. Determine the sum. 1/64 + 1/16 + 1/4 + … 1024 3. Determine the sum. -2 + 4 - 8 … -8192 Sigma Notation (Geometric) Infinite Geometric Series Application of Sequences and Series Arithmetic Series 1. A theater has 32 rows of seats. if there are 26 seats in the first row, 30 in the second, 34 in the 3rd, and so on. How many seats are there in all? 2. There are 20 rows of seats in a concert hall. 25 seats in the first, 27 seats in the second, 29 seats in the third, and so on. If the price per ticket is $2500, how much will be the total sales for a one-night concert if all the seats are sold out? 3. A company sells $160,000 worth of printing paper during its first year. The sales manager has set a goal of increasing annual sales of printing by $250,000 each year for 9 years. Assuming this goal is met, find the total sales of printing paper during the first 10 years this company is in operation. Population Population (increasing) Formula Example 1. The population of a town increases at a rate of 6.5% annually. Its present population is 200,000. What will be the population at the end of 5 years? 2. The population of a city increases at a rate of 5.4% annually. Its present population is at 853,000. What will be the population by the end of 10 years? 3. The number of bacteria in a certain culture doubles every 30 minutes. If there were 200 bacteria present in the culture originally, how many bacteria will be present at the end of the 5th hour? 1+1=2 Population (decreasing) Formula Population (with multiplier) Formula Compound m annually 1 bi/semi-annually/ 2 semestrally quarterly 4 monthly 12 Example 1. Ingrid invested $150,000 in a bank that offers 6% interest compounded annually. How much will her money bt after 10 years? Total Vertical Distance Formula Relations Correspondence in any type of ordered pairs Any set of ordered pairs Types of Relations One to One Many to One One to Many Many to Many One x value One x value Many x values Many x values to one y value to many y to one y value to many y Function values Not a function values Function Not a function Relations as a Set of Ordered Pairs Using the elements of Set A = {1, 2, 3, 4, 5}, construct a set of ordered pairs F that describes the relation “The abscissa is twice the ordinate.” Determine the domain and the range. X = 2y A = {1, 2, 3, 4, 5} B = {½, 1, 3/2, 2, 5/2} F = {(1, ½), (2, 1), (3, 3/2), (4, 2), (5, 5/2)} Relations Described by a Table Domain Range Husband Wife Moses Zipporah Ahab Jezebel Boaz Ruth Abraham Sarah Aquila Priscilla Relations Expressed in an Arrow Diagram {(1, a), (1, b), (2, d), (4, c)} Relations Expressed as an Equation/Formula Given the domain {-2, -1, 0, 1, 2}, determine the set of all possible ordered pairs, and the range of each relation. y = 3x - 2 x -2 -1 0 1 2 {(-2, -8), (-1, -5), (0, -2), (1, 1), (2, 4)} y -8 -5 -2 1 4 Range = {-8, -5, -2, 1, 4} Relations Described Graphically Show graphically the relation “y is equal to one more than the square of x” y = x² + 1 x -2 -1 0 1 2 {(-2, 5), (-1, 2), (0, 1), (1, 2), (2, 5)} y 5 2 1 2 5 * parabola Function A special type of relation in which no two ordered pairs have the same abscissa. A set of ordered pairs (x, y) such that no two ordered pairs have the same x value but different y values. A rule of correspondence between two nonempty sets, such that to each element of the first set, corresponds to one and only one element of the second set. f(x) the notation The value of the function f at the number x where y = f(x) Examples 1. Let f(x) = 3x - 2 f(-2) f(7a) f(x-2) 2. Let g(x) = f(7a) f(x-2)  3. Let f(x) = 2x - 3 f(x-2) f(2x) - f(-x) f( ) Operations on Functions Addition of Subtraction of Multiplication of Division of Functions Functions Functions Functions Add f(x) = 4x - 5 and Subtract f(x) = 4x - 5 Multiply f(x) = 4x - 5 Divide f(x) = 4x - 5 g(x) = 2x and g(x) = 2x and g(x) = 2x and g(x) = 2x (f + g)(x) (f - g)(x) (f * g)(x) = f(x) + g(x) = f(x) - g(x) = f(x) * g(x) = 4x - 5 + 2x = 4x - 5 - 2x = (4x - 5)(2x) = 6x - 5 = 2x - 5 = 8x² - 10x =  = Formulas Arithmetic Sequence Formula Arithmetic Mean Formula Arithmetic Series Formula Sigma Notation (Arithmetic) Geometric Sequence Formula Geometric Mean Formula Geometric Series Formula Sigma Notation (Geometric) Infinite Geometric Series Population (increasing) Population (decreasing) Population (with multiplier) Total Vertical Distance