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 the difference between the sum of the first 10 terms of {a_n} and the sum of the first 10 terms of {b_n}.

Verified step by step guidance

1. Identify the type of sequences {a_n} and {b_n}. For {a_n} = -5, 10, -20, 40, ..., observe the pattern of terms to determine if it is arithmetic or geometric. Similarly, analyze {b_n} = 10, -5, -20, -35, ... to classify its sequence type.
2. For {a_n}, check the ratio between consecutive terms: calculate $\frac{\mathrm{a\_2}}{\mathrm{a\_1}}$, $\frac{\mathrm{a\_3}}{\mathrm{a\_2}}$, etc. If the ratio is constant, {a_n} is geometric. If the difference between terms is constant, it is arithmetic.
3. For {b_n}, calculate the differences between consecutive terms: $a\_2\; -\; a\_1$, $a\_3\; -\; a\_2$, etc. If the difference is constant, {b_n} is arithmetic. Otherwise, check for other patterns.
4. Once the sequence types are identified, use the appropriate formula to find the sum of the first 10 terms. For an arithmetic sequence, use $S\_n\; =\; \backslash frac\{n\}\{2\}\; (2a\_1\; +\; (n-1)d)$, where $a\_1$ is the first term and $d$ is the common difference. For a geometric sequence, use $S\_n\; =\; a\_1\; \backslash frac\{1\; -\; r^n\}\{1\; -\; r\}$ if $r\; \backslash neq\; 1$, where $r$ is the common ratio.
5. Calculate the sum of the first 10 terms for both {a_n} and {b_n} using the formulas, then find the difference by subtracting the sum of {b_n} from the sum of {a_n}.

Key Concepts

Arithmetic and Geometric Sequences

Sequences are ordered lists of numbers following a specific pattern. Arithmetic sequences have a constant difference between terms, while geometric sequences have a constant ratio. Identifying the type of sequence helps determine the formula for the nth term and the sum of terms.

Sum of the First n Terms of a Sequence

The sum of the first n terms of a sequence can be found using specific formulas. For arithmetic sequences, the sum is n/2 times the sum of the first and nth terms. For geometric sequences, the sum depends on the common ratio and uses a different formula. Knowing these formulas is essential to calculate sums efficiently.

Term-by-Term Operations on Sequences

When comparing or combining sequences, operations like subtraction or addition are performed term-by-term. To find the difference between sums of two sequences, calculate each sum separately and then subtract. Understanding this process ensures accurate manipulation of sequences.