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Chapter 11: Problem 16

Find the indicated term of each sequence. $$ a_{n}=(3 n+2)^{2} ; a_{6} $$

Short Answer

Expert verified
The 6th term of the sequence is 400.

Step by step solution

01

Identify the Given Formula and Term

The given formula is for the nth term of the sequence: \[a_n = (3n + 2)^2\]We need to find the 6th term of the sequence, which means we will use n = 6.
02

Substitute n with 6

Replace n with 6 in the formula: \[a_6 = (3 \cdot 6 + 2)^2\]
03

Simplify Inside the Parentheses

Calculate inside the parentheses first: \[3 \cdot 6 = 18\]Then add 2: \[18 + 2 = 20\]So, the expression inside the parentheses becomes 20.
04

Square the Result

Now square the result from the previous step: \[20^2 = 400\]
05

Write the Final Answer

The 6th term of the sequence is 400. So, \[a_6 = 400\]

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Key Concepts

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

nth term
In algebra, sequences often have a specific pattern. The position of a term in the sequence is represented by 'n'. The general form of the term in the sequence is called the 'nth term' and is usually denoted as \(a_n\). For example, in the formula \(a_n = (3n + 2)^2\), every term in the sequence can be found by substituting different values of \(n\). For the 6th term, we replace \(n\) with 6.
substitution
Substitution is a fundamental technique in algebra. It involves replacing a variable with a specific value. In the context of sequences, after identifying the term you are interested in (like the 6th term), you substitute the position number into the general formula. For our sequence formula \(a_n = (3n + 2)^2\), substituting \(n = 6\) gives us: \[a_6 = (3 \times 6 + 2)^2\]. Substitution helps us transform a general formula into a specific calculation.
simplification
Simplification is the process of making an expression easier to manage. After substitution, the next step is simplification. This includes performing arithmetic operations to get a single value. For our example, after substituting \(n = 6\): \[a_6 = (3 \times 6 + 2)^2\], we simplify inside the parentheses: \[3 \times 6 = 18\]. Then add the 2: \[18 + 2 = 20\]. Finally, we square the result: \[20^2 = 400\]. Simplification reduces the complexity of an expression step by step.
arithmetic sequence
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. Although the given example \(a_n = (3n + 2)^2\) is not an arithmetic sequence, understanding arithmetic sequences is crucial. An example of an arithmetic sequence is: \[2, 5, 8, 11, \text{...}\], where the common difference is 3. Arithmetic sequences help us recognize patterns and predict future terms easily. In our example, although the pattern isn't arithmetic, applying arithmetic principles to substitution and simplification aids in correctly solving complex sequences.

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