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 6 terms of {an} and the sum of the infinite seris containing all the terms of {cn}.

Accepted Answer

Identify the type of each sequence. For $$ \{a_n\} = -5, 10, -20, 40, \ldots $$, observe the pattern of terms to determine if it is geometric or arithmetic. Notice the sign changes and the ratio between consecutive terms. Once confirmed that $$ \{a_n\}  is $$geometric, find the common ratio $$ r  by $$dividing the second term by the first term: $$ r = \frac{a_2}{a_1} . $$Calculate the sum of the first 6 terms of $$ \{a_n\} $$ using the formula for the sum of the first $$ n $$ terms of a geometric sequence: $$ S_n = a_1 \cdot \frac{1 - r^n}{1 - r} $$, where $$ a_1  is $$the first term and $$ r  is $$the common ratio. Analyze the sequence $$ \{c_n\} = -2, 1, -\frac{1}{2}, \frac{1}{4}, \ldots  to $$confirm it is geometric by checking the ratio between consecutive terms. Then, find the common ratio $$ r_c . $$Since $$ \{c_n\}  is an $$infinite geometric series with $$ |r_c| \u003c 1 $$, find its sum using the infinite geometric series sum formula: $$ S_\infty = \frac{a_1}{1 - r_c} . $$Finally, find the difference between the sum of the first 6 terms of $$ \{a_n\} $$ and the sum of the infinite series $$ \{c_n\}  by $$subtracting the two sums.

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 6 terms of {a_n} and the sum of the infinite seris containing all the terms of {c_n}.

Key Concepts

Arithmetic and Geometric Sequences

Sum of a Finite Number of Terms

Sum of an Infinite Geometric Series