College Algebra: Sequence and Series Problem Solutions

This document provides detailed solutions to a series of problems related to sequences and series. The problems cover both arithmetic and geometric progressions, including finding explicit formulas for the general term (an) and calculating specific terms within a sequence. The solution includes finding the sum of a finite number of terms (Sn) and the sum of an infinite number of terms (S∞) where applicable. The problems also involve real-world applications, such as calculating the price of a cookbook over several years and determining the growth of an influencer's followers. The solutions presented include all the steps, formulas, and calculations required to arrive at the final answers.

Problem 1: For the sequence $76.19, $75.62, $75.05, $74.48, …….
2A. Find the explicit formula (general rule) for the general term an of the sequence given above. and Use
your explicit formula to find the 2001st term of the given sequence.
Ans)
a2−a1=−$ 0.57
a3−a2 =−$ 0.57
a4−a3=−$ 0.57
Therefore it has a common difference d=−$ 0.57
So it is an Arithmetic Progression,
an=a1 + ( n−1 ) d
a2001=a1+ 2000× (−$ 0.57 )=$ 76.19−$ 1140=−$ 1063.81
2B. Find S2001 which is the sum of the first 2001 terms of the given sequence.
Ans)
Sn= n
2 [2 a1 + ( n−1 ) d ]
S2001=2001
2 [2 ( $ 76.19 ) + ( 2000 ) (−$ 0.57 ) ]=−$ 988,113.81
Problem 2. For the sequence 12000, 4800, 1920, 768, ….
2A. Find the explicit formula (general rule) for the general term an of the sequence given above and
calculate the 10th term of sequence.
Ans)
a2
a1
=0.4
a3
a2
=0.4

a4
a3
=0.4
Therefore, it is a Geometric Progression, r = 0.4
So,
an=a1 (0.4)n−1
a10=12000(0.4)9 =3.146
2B. Find S∞, which is the sum of an infinite number of terms of the sequence.
Ans)
The sum of till nth term is given by,
Sn=a1 (1− ( 0.4 )n )/(1−0.4)
When n tends to infinity it becomes
S∞ = a1
1−0.4 = 12000
0.6 =20000
Problem 3. Starting this year in 2019, an annual spring cookbook will be published each year in May. The
cookbook costs $13.75 in May 2019, and then each year the price will increase by $ .94 over the
previous year’s price.
3A. Find the explicit formula (general rule) for the sequence of the cookbook prices each year. The first
term a1 should represent the price in the year May 2019 and Use the explicit formula to determine what
the price will be in May 2042.
Ans)
Therefore it has a common difference d=$ 0. 94
So it is an Arithmetic Progression,
an=a1 + ( n−1 ) d
First term=a1=a2019=$ 13.75
a2 4=a2 042=a2019+2 4 × ( $ 0. 94 )=$ 13.75+ $ 22.56=$ 36.3 1
3B. Use a sequence formula to determine how much you will have spent in total if you buy a cookbook
each year from May 2019 to May 2042.

Ans)
Sn= n
2 [2 a1 + ( n−1 ) d ]
S24= 2 4
2 [ 2 ( $ 13.75 )+ ( 23 ) ( $ 0.94 ) ]=$ 589.44 .
Problem 4. On Jan. 1, 2019, an influencer had 500,000 followers on Instagram. On Feb 1, 2019, he had
12.5% more followers than he had in January. With marketing, the influencer wants to continue to
increase by 12.5% each month over the previous month.
4A. Find the explicit formula (general rule) for a sequence of the number of followers he hopes to have
on the first day of each month, starting with a1 representing the amount in Jan 1, 2019.
Ans)
Therefore, it is a Geometric Progression with r = 0.125
So,
an=a1 (0.125)n−1=500000 (0.125)n−1
Where n is the number of months
4B. According to this sequence how many followers will the influencer have in January 2021
Ans)
The sum of till nth term is given by,
Sn=a1 (1− ( 0.125 ) n)/(1−0.125)
Now n = 120 as 10 years and for each month
S120= 500000 ( 1− ( 0.125 ) 120 )
1−0.125 =571428