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Intermediate Algebra – Chapter 11 Summary
Sequences and Series
1. There is a well-known story about Karl Friedrich Gauss when he was in elementary
school. His teacher got mad at the class and told them to add the numbers 1 to 100
and give him the answer by the end of the class. About 30 seconds later, Gauss gave
the teacher the answer. What was the answer? How did little Gauss figure it out so
fast?
2. Suppose that the teacher asked Gauss to add the first 100 even natural numbers,
what do you think his solution will be?
3. Suppose that the teacher asked Gauss to add the first 100 odd natural numbers,
what do you think his solution will be?
Int. Algebra_Ch. 11_Summary
pg 1 of 13
4. Suppose that the teacher asked Gauss to add the first 100 terms of each of the
following sequences, what do you think his solution will be?
a)
b)
c)
d)
e)
f)
3, 6, 9, 12, ...
5, 3, 1, -1, -3, ...
7, 7 + d, 7 + 2d, 7 + 3d, ...
a, a + d, a + 2d, a + 3d, ...
Challenge: 1, 2, 4, 8, 16,...
Super challenge: 1, 4, 9, 16, 25, ...
Int. Algebra_Ch. 11_Summary
pg 2 of 13
Arithmetic Sequences and Series
Example
Explain the pattern in the following sequence
1, 4, 7, 10, 13, ...
Use the pattern to find the next two terms.
1, 4, 7, 10, 13,_______, _______
Find the value of the 100th term. Explain.
Graph the sequence. What do you notice about the graph?
Find the sum of the first 100 terms.
Int. Algebra_Ch. 11_Summary
pg 3 of 13
Arithmetic Sequences and Series
A sequence is an arithmetic sequence if
Formulas for finding the value of each term in an arithmetic sequence:
a)
Recursive formula:
b)
Explicit formula: (dependent only on the term position, n)
Graph of an arithmetic sequence:
Sum of a finite arithmetic series:
Sum of, Sn, of the first n terms of an arithmetic series with first term, t1, and the nth term, tn, is:
Int. Algebra_Ch. 11_Summary
pg 4 of 13
More Practice:
1. The second term of an arithmetic sequence is 11; its fifth term is 32.
a) Find the first term.
b) Write a recursive formula and an explicit formula for sequence.
c) Find S20.
2. Find the arithmetic means (missing terms) of the sequence:
2, _______, ________, ________,16
2.
5
∑3i + 2 =
i=1
3.
100
∑4−i =
i=5
4. Write the following series using summation notation as in #2 &3:
a) 3 + 5 + 7 + ... + 101
b) 1 + 4 + 9 + 16 ... + 100
Int. Algebra_Ch. 11_Summary
pg 5 of 13
5. A radio station considered giving $4,000 every day in the month of August for a total of $124,000. Instead,
they decided to increase the amount given away every day while still giving away the same total amount. If
they want to increase the amount by $100 each day, how much should they give away the first day?
6. Tickets for a certain show were printed bearing numbers from 1 to 100. Odd number tickets were sold by
receiving cents equal to thrice (three times) of the number on the ticket while even number tickets were
issued by receiving cents equal to twice of the number on the ticket. How much amount was received by the
issuing agency?
7. A factory owner repays his loan of $208,800 by $20,000 in the first monthly installment and then increases
the payment by $1,000 in every installment. In how many installments he will clear his loan?
8. True or false. Explain.
a) Doubling each term in an arithmetic series will double the sum.
b) Doubling the number of terms in an arithmetic series, but keeping the first term and common difference
the same, will double the sum.
Int. Algebra_Ch. 11_Summary
pg 6 of 13
Geometric Sequences and Series
Example
Explain the pattern in the following sequence
1, 3, 9, 27, 81, ...
Use the pattern to find the next two terms.
1, 3, 9, 27, 81,_______, _______
Find the value of the 100th term. Explain.
Graph the sequence. What do you notice about the graph?
Find the sum of the first 100 terms.
Int. Algebra_Ch. 11_Summary
pg 7 of 13
Geometric Sequences and Series
A sequence is a geometric sequence if
Formulas for finding the value of each term in a geometric sequence:
a)
Recursive formula:
b)
Explicit formula: (dependent only on the term position, n)
Graph of a geometric sequence:
Sum of a finite geometric series:
Sum of, Sn, of the first n terms of a geometric series with first term, t1, and the nth term, tn, is:
Int. Algebra_Ch. 11_Summary
pg 8 of 13
Infinite Geometric Series:
1. Find the sum of the infinite sequence 1+ 2 + 4 + 8 + 16, ....
2. Find the sum of the infinite sequence
1 1 1 1
+ + + ,...
2 4 8 16
Sum of an infinite geometric series:
Sum of, Sn, of the first n terms of a geometric series with first term t1 and common ratio r (r ≠ 1):
Int. Algebra_Ch. 11_Summary
pg 9 of 13
More Practice:
1. Suppose a ball is dropped from a height of 10 feet, and each time it falls, it rebounds to 80% of the height
from which it fell.
a) Find the height of the ball after five rebounds.
b) Find the total distance the ball has traveled after 5 rebounds.
c) Assume that the ball can rebound an infinite number of times, find the total distance that the ball travels.
Int. Algebra_Ch. 11_Summary
pg 10 of 13
2. Annuity:
a) Annuity is a sequence of equal payments made at equal time periods. Suppose that you start saving for your
retirement at the age of 20 by depositing $100 at the end of each month into an annuity that pays 5% interest
per year compounded monthly.
By the time you retire at the age of 65, what would be the total balance in the account when you retire?
b) If P dollars is deposited into an account at the end of each compounding period at an annual interest rate, r,
compounded n times per year, find a formula that represents the total value of the annuity, A, after t years.
c) Suppose that you want to retire with a million dollars. What is the minimum amount that must be deposited
at the end of each month into an account that pays 5% interest compounded monthly? Assume that you start
saving at the age of 20 and retire at age 65.
Int. Algebra_Ch. 11_Summary
pg 11 of 13
3. The design shown below, which approximates a fractal, is created with midsegments. Beginning with any
triangle, shade the triangle formed by the three midsegments. Continue the process for each unshaded
triangle. Suppose the perimeter of the original triangle is 1. Fill in the table for the total perimeter of all the
triangles that are shaded for each stage.
Stage 0
Stage
Total
perimeter of
all shaded
triangles
Stage 1
1
Stage 2
2
Stage 3
3
Stage 4
4
5
Find a formula for the total perimeter of all shaded triangles at any stage n.
Suppose the area of the original triangle is 1. Fill in the table for the total area of all the triangles that are
shaded for each stage.
Stage
Total area of
all shaded
triangles
1
2
3
4
5
Find a formula for the total area of all shaded triangles at any stage n.
Int. Algebra_Ch. 11_Summary
pg 12 of 13
4.
Joe started a pyramid scheme. He asks Molly for a $1 with the promise that if Molly convinces two people
to join and pay the entrance fee of $1, Molly will get $2 which would double up her initial investment.
The other two people will have to recruit two people each in order to double up their money, and the
pattern continues.
a) How many people are involved in the scheme after 3 levels? 10 levels? n levels?
b) If the pyramid scheme ends at level 3, how many people will lose their money? What if it ends at
level 10?
c) Why would such scheme be considered fraudulent and illegal?
5.
In The 8-Ball Model Pyramid Scheme, the person recruiting does not get paid at all until they
have recruited 3 levels worth of new members. Thus Person A at level 1 recruits 2 people
at level 2, these 2 recruit 4 at level 3, and these 4 recruit 8 at level 4. When the 8 are
recruited, Person A receives the "participation fee" for all 8 people of level 4. If the fee was
$1000, then Person A would receive $8000. If 16 people are then recruited at level 5, then
the 2 people at level 2 would each receive $8000.
What percent of the members would be losing their money at any level in this scam?
Int. Algebra_Ch. 11_Summary
pg 13 of 13