Problem 3 - Beginning and Intermediate Algebra

Question

Fill in each blank with the correct response. If \(a_{n}=2 n,\) then \(a_{40}=\) ____.

Step-by-Step Solution

Answer

80

Step 1 - Understand the Sequence

Identify the given sequence formula: The general formula for the sequence is given as \(a_{n} = 2n\).

Step 2 - Identify the Term to Find

Determine which term of the sequence we need to find: In this case, we are asked to find \(a_{40}\), which means we need to substitute \(n = 40\) into the formula.

Step 3 - Substitute and Calculate

Substitute \(n = 40\) into the sequence formula and calculate the result: \[a_{40} = 2 \times 40 = 80\]

Key Concepts

sequence formula

Sequences are an ordered list of numbers following a specific pattern. To describe any sequence, we use a sequence formula. This formula expresses the nth term of the sequence in terms of 'n', which represents the position in the sequence. For our exercise, the sequence formula is given as: \[a_n = 2n\] This means to find any term in the sequence, multiply the term’s position (n) by 2. For example, for the first term ( = 1\textendash a_1), you benefit to apply: \[a_1 = 2 \times 1 = 2\] Therefore, each term can be computed using this simple multiplication rule.

substitution

Once you understand the sequence formula, identifying the specific term to find is essential. In our case, we're asked to compute the 40th term: = 40. To do this, substitute 40 for 'n' in the sequence formula. The substitution process involves replacing 'n' with 40: \[a_{40} = 2 \times 40\] Substituting correctly is crucial to finding the right answer. Remember, we substitute 'n' with the desired term's position to see its value.

calculation

After substituting the value into the formula, the next step is to perform the calculation. For our example, substitute 40 into our sequence formula: \[a_{40} = 2 \times 40 = 80\] Breaking it down: \[2 \times 40 = 80\] Means we multiply 2 by 40 to get 80. Performing these calculations ensures you find the correct sequence term. Always verify your calculation to ensure accuracy. For instance, repeatedly multiplying 2 by appropriate 'n' values ensures understanding and accuracy when working with sequences.