Find the nth term of each sequence. a) 11, 8, 5, 2, -1

Understand the Problem

The question is asking to find the nth term of the given sequence: 11, 8, 5, 2, -1. This involves identifying the pattern of the sequence and formulating an expression to calculate any term in the sequence based on its position (n).

Answer

The nth term of the sequence is given by $a_n = 14 - 3n$.

Steps to Solve

Identify the pattern in the sequence

We observe the given sequence: 11, 8, 5, 2, -1. The sequence is decreasing. Let's find the difference between consecutive terms:

$8 - 11 = -3$
$5 - 8 = -3$
$2 - 5 = -3$
$-1 - 2 = -3$

The common difference is $-3$.

Formulate the nth term of the sequence

The formula for the nth term ($a_n$) of an arithmetic sequence can be expressed as:

$$ a_n = a_1 + (n - 1) \cdot d $$

Where:

$a_1$ is the first term (11 in this case),
$d$ is the common difference (-3).

Substituting values:

$$ a_n = 11 + (n - 1) \cdot (-3) $$

Simplify the formula

Now we simplify the expression:

$$ a_n = 11 - 3(n - 1) $$

Expanding this gives:

$$ a_n = 11 - 3n + 3 $$

Combining like terms:

$$ a_n = 14 - 3n $$

The nth term of the sequence is given by the formula:

$$ a_n = 14 - 3n $$

More Information

This formula allows you to find any term in the sequence by substituting the value of ( n ). For instance, when ( n = 1 ), ( a_1 = 14 - 3(1) = 11 ), which matches the first term of the sequence.

Tips

Forgetting the common difference: Ensure to consistently calculate the difference between terms.
Misapplying the arithmetic sequence formula: Make sure to properly plug in the first term and the common difference in the formula.

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