Problem 34 - Algebra and Trigonometry

Question

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 8th term is 4; 18th term is - 96

Step-by-Step Solution

Answer

The first term is 74 and the common difference is -10. Recursive formula: a_1 = 74, a_n = a_(n-1) -10. nth term formula: a_n = 84 - 10n.

Step 1: Determine the Common Difference

In an arithmetic sequence, the common difference (d) can be found using the formula for the nth term: a_n = a_1 + (n-1)dWhere a_n is the nth term, a_1 is the first term, and d is the common difference. We have two terms given in the sequence: a_8 = 4 and a_18 = -96. Set up two equations to solve for d:a_8: 4 = a_1 + 7d a_18: -96 = a_1 + 17d

Step 2: Solve for the Common Difference

Subtract the first equation from the second equation to eliminate a_1:a_18 - a_8 = (a_1 + 17d) - (a_1 + 7d) -96 - 4 = a_1 + 17d - a_1 - 7d -100 = 10d d = -10

Step 3: Find the First Term

Now use the common difference (d = -10) to find the first term (a_1). Substitute d into one of the original equations:a_8: 4 = a_1 + 7(-10) 4 = a_1 - 70 a_1 = 74

Step 4: Find the Recursive Formula

The recursive formula for an arithmetic sequence is shown by:a_1 = first term a_n = a_(n-1) + d a_1 = 74 a_n = a_(n-1) -10

Step 5: Find the Formula for the nth Term

Use the formula for the nth term of an arithmetic sequence: a_n = a_1 + (n-1)d a_n = 74 + (n-1)(-10) a_n = 74 - 10n + 10 a_n = 84 - 10n

Key Concepts

Common Difference

In an arithmetic sequence, the 'common difference' is the difference between consecutive terms. Think of it as the amount each term goes up or down from the previous one.
We can find this difference using the formula for the nth term: \(a_n = a_1 + (n-1)d\), where \(a_n\) is the nth term, \(a_1\) is the first term, and \(d\) is the common difference.
For example, if the 8th term is 4 and the 18th term is -96, we can set up two equations to find \(d\).
Using the given terms:

• \(4 = a_1 + 7d\) (for the 8th term)
• \(-96 = a_1 + 17d\) (for the 18th term)

Subtract these equations to eliminate the first term (\(a_1\)), giving us:
\[-100 = 10d\]
Solving this, we get:\[d = -10\]

Recursive Formula

A recursive formula is a way to define each term of a sequence using previous terms. This is useful for generating the sequence step-by-step.
An arithmetic sequence can be expressed recursively as:

• First term: \(a_1 = \text{first term}\)
• Subsequent terms: \(a_n = a_{n-1} + d\)

In our example, the first term \(a_1\) is 74 and the common difference \(d\) is -10. Thus, the recursive formula becomes:
\[a_1 = 74\]
\[a_n = a_{n-1} - 10\]
This shows that each term is 10 less than the previous term.

nth Term Formula

The nth term formula directly calculates any term in an arithmetic sequence without needing to know the previous term.
The nth term formula is given by:
\[a_n = a_1 + (n-1)d\]
Substituting the first term and common difference, we have:
\[a_n = 74 + (n-1)(-10)\]
Simplifying this, we get:
\[a_n = 74 - 10n + 10\]
\[a_n = 84 - 10n\]
This formula lets you find any term in the sequence by just plugging in the value of \(n\).

Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence.
If we want to sum the first \(n\) terms, we can use the formula:
\[S_n = \frac{n}{2}(a_1 + a_n)\]
Here, \(S_n\) is the sum of the first \(n\) terms, \(a_1\) is the first term, and \(a_n\) is the nth term.
For our example sequence with \(a_1 = 74\) and \(d = -10\), calculating the 8th term using the nth term formula gives us 4 (already known).
To sum the first 8 terms:
\[S_8 = \frac{8}{2}(74 + 4)\]
\[S_8 = 4 \times 78 = 312\]
This approach quickly adds up terms in an arithmetic sequence.