Problem 30 - Algebra and Trigonometry

Question

\(27-36\) . Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence. $$ 11,8,5,2, \dots $$

Step-by-Step Solution

Answer

Common difference is -3, fifth term is -1, nth term is \(14-3n\), 100th term is -286.

Step 1: Identify the First Term and Common Difference

The first term of the sequence is 11. To find the common difference, subtract the second term from the first term: \[ d = 8 - 11 = -3 \] Thus, the common difference is \(-3\).

Step 2: Calculate the nth Term Formula

The formula for the nth term of an arithmetic sequence is given by: \[ a_n = a_1 + (n-1) imes d \] Substituting the known values, \( a_1 = 11 \) and \( d = -3 \), we get: \[ a_n = 11 + (n-1)(-3) \] This simplifies to: \[ a_n = 11 - 3n + 3 = 14 - 3n \] Thus, the nth term can be calculated using \( a_n = 14 - 3n \).

Step 3: Determine the Fifth Term

Substitute \( n = 5 \) into the nth term formula to find the fifth term:\[ a_5 = 14 - 3 imes 5 \] Calculating gives:\[ a_5 = 14 - 15 = -1 \] So, the fifth term of the sequence is \(-1\).

Step 4: Calculate the 100th Term

Substitute \( n = 100 \) into the nth term formula to find the 100th term:\[ a_{100} = 14 - 3 imes 100 \] Calculating gives:\[ a_{100} = 14 - 300 = -286 \] Thus, the 100th term of the sequence is \(-286\).

Key Concepts

Common Difference

The common difference is a key concept in arithmetic sequences. It is the constant amount that each term increases or decreases by when moving from one term to the next.

• To find the common difference, simply subtract the first term from the second term in the sequence.
• In our sequence 11, 8, 5, 2, ..., subtract 11 from 8 which gives a common difference of \(-3\).
• This indicates that each term in the sequence is 3 less than the previous term.

Understanding the common difference helps us quickly identify how the sequence is progressing. It sets the foundation for developing the nth term formula.

Nth Term Formula

The nth term formula is used to find any term in an arithmetic sequence without listing all previous terms. The formula is given by: \[ a_n = a_1 + (n-1) \times d \]Here:

• \( a_n \) is the nth term we are trying to find.
• \( a_1 \) is the first term of the sequence, which is 11 in our sequence.
• \( d \) is the common difference, which we have found is \(-3\).

By substituting these values, we develop a simplified formula: \[ a_n = 11 + (n-1)(-3) = 14 - 3n \]This simplified formula allows us to plug in any value of \( n \) to find a specific term quickly.

Fifth Term

Calculating specific terms like the fifth term is straightforward once we have the nth term formula. Simply substitute \( n = 5 \) into the nth term formula:\[ a_5 = 14 - 3 \times 5 \]By performing the calculation:

• Multiply 3 by 5 to get 15
• Subtract 15 from 14, resulting in \(-1\).

Thus, the fifth term of the sequence is \(-1\). Recognizing this direct calculation method not only confirms our sequence pattern but also underscores the power of the nth term formula.

100th Term

Finding something like the 100th term without listing all previous terms showcases the efficiency of the nth term formula. Use the formula:\[ a_{100} = 14 - 3 \times 100 \]Here's how the calculation unfolds:

• Multiply 3 by 100, giving us 300.
• Subtract 300 from 14, arriving at \(-286\).

So, the 100th term in our arithmetic sequence is \(-286\). This approach highlights how arithmetic sequences, with defined formulas and patterns, can make seemingly challenging tasks straightforward.