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Chapter 11: Problem 14

Find the indicated term of the given sequence. $$a_{n}=(-1)^{n-1}(4.6 n-18.3) ; a_{12}$$

Short Answer

Expert verified
a_{12} = -36.9

Step by step solution

01

Identify the given term

We are given the sequence formula: a_{n} = (-1)^{n-1}(4.6n-18.3). We need to find the value of a_{12}.
02

Substitute the value of n

To find a_{12}, substitute n = 12 into the formula: a_{12} = (-1)^{12-1}(4.6(12) - 18.3).
03

Simplify the exponent

Simplify the exponent first: (-1)^{12-1} = (-1)^{11}. Since the exponent is odd, (-1)^{11} = -1.
04

Calculate the expression inside the parentheses

Next, compute the expression inside the parentheses: 4.6(12) - 18.3 = 55.2 - 18.3 = 36.9.
05

Multiply by the simplified exponent

Finally, multiply by the value obtained from the exponent: a_{12} = -1 * 36.9 = -36.9.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

arithmetic sequence
In mathematics, a sequence is an ordered list of numbers. An arithmetic sequence is a type of sequence where the difference between consecutive terms is constant. This difference is called the 'common difference' (d).
For example, in the sequence 2, 5, 8, 11, ..., the common difference is 3.
Understanding arithmetic sequences is crucial in solving many math problems, including ones that involve patterns and trends.
term calculation
To find a specific term in a sequence, we need to employ the given sequence formula and substitute the necessary values.
For instance, in the sequence given by the formula \(a_{n}=(-1)^{n-1}(4.6n-18.3)\), 'n' represents the term number.
To find the 12th term (\

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Most popular questions from this chapter

Prove that $$ \left(\begin{array}{c} n \\ k-1 \end{array}\right)+\left(\begin{array}{l} n \\ k \end{array}\right)=\left(\begin{array}{c} n+1 \\ k \end{array}\right) $$ for any natural numbers \(n\) and \(k, k \leq n\)

The sides of a square are \(16 \mathrm{cm}\) long. A second square is inscribed by joining the midpoints of the sides, successively. In the second square, we repeat the process, inscribing a third square. If this process is continued indefinitely, what is the sum of all the areas of all the squares? (Hint: Use an infinite geometric series.) (PICTURE CANNOT COPY)

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