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 sequences {a_n} and {b_n} and determine their types (arithmetic or geometric). For {a_n} = -5, 10, -20, 40, ..., observe the pattern of terms and check if there is a common ratio or common difference.
2. For {a_n}, calculate the common ratio \( r_a \) by dividing the second term by the first term: \( r_a = \frac{10}{-5} \). Verify if this ratio holds for subsequent terms to confirm it is geometric.
3. Similarly, analyze the sequence {b_n} = 10, -5, -20, -35, ... to determine if it is arithmetic or geometric. Calculate the differences between consecutive terms to check for a common difference \( d_b \).
4. Once the nature of both sequences is confirmed, use the appropriate formula to find the sum of the first 10 terms for each sequence. For a geometric sequence, use \( S_n = a_1 \frac{1 - r^n}{1 - r} \) where \( a_1 \) is the first term and \( r \) is the common ratio. For an arithmetic sequence, use \( S_n = \frac{n}{2} (2a_1 + (n-1)d) \) where \( d \) is the common difference.
5. Calculate the difference between the sum of the first 10 terms of {a_n} and the sum of the first 10 terms of {b_n} by subtracting \( S_{10}^{b} \) from \( S_{10}^{a} \).

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 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, use S_n = n/2 (first term + last term). For geometric sequences, use S_n = a(1 - r^n)/(1 - r), where a is the first term and r is the common ratio.

Term-by-Term Operations on Sequences

When comparing sums of sequences, understanding how to perform operations like subtraction between sums is essential. This involves calculating each sum separately and then finding their difference, ensuring careful handling of signs and sequence types.