Problem 52 - Contemporary Precalculus

Question

Find the third and the sixth partial sums of the sequence. $$\left\\{\left(2 n-3 n^{2}\right)^{2}\right\\}$$

Step-by-Step Solution

Answer

Answer: The third partial sum is 506, and the sixth partial sum is 15547.

Step 1: Find the first three terms of the sequence

To find the first three terms of the sequence, substitute n=1, n=2, and n=3 in the expression \((2n-3n^2)^2\): Term 1: \((2(1)-3(1)^2)^2 = (2-3)^2 = 1\) Term 2: \((2(2)-3(2)^2)^2 = (4-12)^2 = (-8)^2 = 64\) Term 3: \((2(3)-3(3)^2)^2 = (6-27)^2 = (-21)^2 = 441\) So, the first three terms are 1, 64, and 441.

Step 2: Find the third partial sum

To find the third partial sum, add the first three terms of the sequence: Third partial sum = 1 + 64 + 441 = 506

Step 3: Find the next three terms of the sequence

To find the next three terms of the sequence (terms 4, 5, and 6), substitute n=4, n=5, and n=6 in the expression \((2n-3n^2)^2\): Term 4: \((2(4)-3(4)^2)^2 = (8-48)^2 = (-40)^2 = 1600\) Term 5: \((2(5)-3(5)^2)^2 = (10-75)^2 = (-65)^2 = 4225\) Term 6: \((2(6)-3(6)^2)^2 = (12-108)^2 = (-96)^2 = 9216\) So, the next three terms are 1600, 4225, and 9216.

Step 4: Find the sixth partial sum

To find the sixth partial sum, add the first six terms of the sequence: Sixth partial sum = 1 + 64 + 441 + 1600 + 4225 + 9216 = 15547 So, the third partial sum is 506 and the sixth partial sum is 15547.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant. This constant is known as the "common difference." For example, in the sequence 2, 5, 8, 11, each term increases by 3, which is the common difference.

To identify or describe an arithmetic sequence, you can use the following properties:

• The first term, which is often denoted as \(a_1\).
• The common difference, denoted by \(d\), which is simply the difference between any two successive terms.

The general formula for any given term \(n\) in an arithmetic sequence is given by: \[ a_n = a_1 + (n-1) imes d \]

Mathematical Series

A mathematical series is the sum of the terms of a sequence. When dealing with arithmetic sequences, summing up the terms, we get an arithmetic series. The sum of the first \(n\) terms of an arithmetic series is represented as the "partial sum."

• A partial sum is a segment of the entire series.
• It shows the accumulated total of the terms up to a certain point.

The formula to find the sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is:\[ S_n = \frac{n}{2} \times (a_1 + a_n) \] where \(a_1\) is the first term and \(a_n\) is the \(n\)-th term. This formula is instrumental when dealing with sequences to calculate various sums systematically.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operation symbols. They are fundamental in forming equations and functions. An expression like \((2n - 3n^2)^2\) involves:

• Variables: Letters that represent numbers, in this case, \(n\).
• Constants: Numbers that have a fixed value, like 2 and 3 in our expression.
• Operations: Addition, subtraction, multiplication, or powers, as seen in the squaring of the expression.

An important part of working with algebraic expressions is learning how to properly substitute numbers for variables and simplify the results, as demonstrated in calculating the terms of a sequence. This process requires careful following of the order of operations (often remembered as PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Understanding and manipulating algebraic expressions is key in solving problems involving sequences and series.