Question

Let {an} = - 5, 10, - 20, 40, ..., {bn} = 10, - 5, - 20, - 35, ..., {cn} = - 2, 1, - 1/2, 1/4 Find the difference between the sum of the first 10 terms of {an} and the sum of the first 10 terms of {bn}.

Accepted Answer

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. 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. 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 $$ d_b . $$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. 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 $$ S_{10}^{a} .$$

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}.

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Concetti chiave

Arithmetic and Geometric Sequences

Sum of the First n Terms of a Sequence

Term-by-Term Operations on Sequences