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Geometric Sequences I
Problem 223: Consider the sequence a. What is the common ration? b. What are the next five terms in the sequence? Solution: A geometric sequence is different from an arithmetic sequence. In an arithmetic sequence, we look for a common difference between the terms in the sequence whereas in the geometric sequence we look for…
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Arithmetic Sequence III
Problem 222: Given . a. Find the closed form: b. Use the closed form to find . Solution: a. The term of an arithmetic sequence is given by (1) where is called the closed form, is the first term in the sequence and is the common difference. To find the common difference, we just…
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Arithmetic Sequence II
Problem 221: Find the term in this sequence: Solution: The term of an arithmetic sequence is given by (1) where is the first term in the sequence and is the common difference. To find the common difference, we just take the difference between the terms. (2) This means that . We are looking…
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Arithmetic Sequences I
Problem 220: Consider the sequence . Find a. the common difference and b. the next five terms of the sequence. Solution: This is an arithmetic sequence. A sequence of numbers that are connected by a common difference. a. The common difference is 7 because (1) b. We can use the common difference to find…
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U-substitution on Definite Integral
Problem 219: Compute (1) Solution: We can solve this integral problem by u-substitution. Since this is a definite integral, we need to change the limits of integration when we make the variable substitution. That is, let so that , and when (2) Hence, now our integral is in term of with the limits…
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Factoring Trinomials Using the “ac” Method
Problem 218: Factor the following polynomials completely: (1) Solution: In order to factor a quadratic polynomial, we need to identify the form of the polynomial. The polynomial is of the form where . The number in front of the squared term is bigger than 1. Thus we will use the “ac” method to factor…
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Linear Inequalities Applications
Problem 217: Alan is loading a pallet with boxes that each weighs 45 pounds. The pallet can safely support no more than 900 pounds. How many boxes can he safely load onto the pallet? Solution: Let be the amount of boxes that Alan can load into the pallet. Then Allen can safely load no more…
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3X3 System of Linear Equation
Problem 216: Solve (1) Solution: We can solve the system of linear equation in many ways. We will do it using matrix. That is, (2) We can now write an augmented matrix using A and b. After using row operations we get (3) Going from bottom to the top we get (4)…
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Simplifying Rational Expressions Using Factoring
Problem 215: Simplify (1) Solution: In order to simplify the rational expression, we need to factor the numerator and denominator. That is, Notice that the numerator and denominator are quadratic equation with leading coefficient 1. Thus (2) That is, (3)
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Using Factoring to Simplify Rational Expressions
Problem 214: Simplify (1) Solution: In order to simplify the rational expression, we need to factor the numerator and denominator. That is, (2) Notice that the numerator is a perfect square which we can factor easily. The denominator is quadratic equation with leading coefficient 1. Thus (3) That is, (4)
One Problem at a Time
