In Exercises 57–62, let {a_n} = - 5, 10, - 20, 40, ..., {b_n} = 10, - 5, - 20, - 35, ..., {c_n} = - 2, 1, - 1/2, 1/4 Find a₁₀ + b₁₀.

Verified step by step guidance

1. Identify the type of sequences given for {a_n} and {b_n}. Check if they are arithmetic or geometric by examining the pattern of terms.
2. For sequence {a_n} = -5, 10, -20, 40, ..., observe the pattern of signs and values. Determine the common ratio if it is geometric by dividing the second term by the first term: $\frac{10}{-}$. Confirm this ratio for the next terms.
3. Once confirmed that {a_n} is geometric, use the geometric sequence formula for the nth term: $a\_n\; =\; a\_1\; \backslash cdot\; r^\{n-1\}$, where $a\_1$ is the first term and $r$ is the common ratio. Calculate the expression for $a\_\{10\}$.
4. For sequence {b_n} = 10, -5, -20, -35, ..., check if it is arithmetic by finding the difference between consecutive terms: $b\_2\; -\; b\_1$, $b\_3\; -\; b\_2$, etc. If the difference is constant, use the arithmetic sequence formula: $b\_n\; =\; b\_1\; +\; (n-1)d$, where $d$ is the common difference. Calculate the expression for $b\_\{10\}$.
5. Add the expressions for $a\_\{10\}$ and $b\_\{10\}$ to find $a\_\{10\}\; +\; b\_\{10\}$. This sum will be the final expression representing the solution.

Key Concepts

Sequences and Terms

A sequence is an ordered list of numbers defined by a specific pattern or rule. Each number in the sequence is called a term, denoted as a_n for the nth term. Understanding how to identify or express terms is essential for finding specific values like a10 or b10.

Arithmetic and Geometric Sequences

Arithmetic sequences have a constant difference between consecutive terms, while geometric sequences have a constant ratio. Recognizing the type of sequence helps determine the formula for the nth term, which is necessary to calculate terms like a10 and b10.

Finding the nth Term and Summation

To find a specific term such as a10 or b10, you must derive or use the nth term formula. Once the terms are found, operations like addition (a10 + b10) can be performed. This involves algebraic manipulation and substitution of n=10 into the formulas.