1. The sequence is a geometric sequence with first term a1 = 3 and ratio r = 3. The specific formula is an = 3(3)n-1. The sum of the first 20 terms is 1743392200 and the infinite sum is 3072.
2. The sequence is a geometric sequence with first term a1 = 1/90 and ratio r = 3. The specific formula is an = (3/90)(3)n-1. The sum of the first 20 terms is 174339220/9 and the infinite sum is 1/30.
3. The sequence results from monthly deposits of $1000 at 1% interest per month. The specific formula is an = 1000
MS Report, When we talked about the conic section it involves a double-napped cone and a plane. If a plane intersects a double right circular cone, we get two-dimensional curves of different types. These curves are what we called the conic section.
MS Report, When we talked about the conic section it involves a double-napped cone and a plane. If a plane intersects a double right circular cone, we get two-dimensional curves of different types. These curves are what we called the conic section.
This will help you in evaluating summation notation.
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This will help you in evaluating summation notation.
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Comparative analysis of x^3+y^3=z^3 and x^2+y^2=z^2 in the Interconnected Sets Vladimir Godovalov
This paper introduces an innovative technique of study z^3-x^3=y^3 on the subject of its insolvability in integers. Technique starts from building the interconnected, third degree sets: A3={a_n│a_n=n^3,n∈N}, B3={b_n│b_n=a_(n+1)-a_n }, C3={c_n│c_n=b_(n+1)-b_n } and P3={6} wherefrom we get a_n and b_n expressed as figurate polynomials of third degree, a new finding in mathematics. This approach and the results allow us to investigate equation z^3-x^3=y in these interconnected sets A3 and B3, where z^3∧x^3∈A3, y∈B3. Further, in conjunction with the new Method of Ratio Comparison of Summands and Pascal’s rule, we finally prove inability of y=y^3. After we test the technique, applying the same approach to z^2-x^2=y where we get family of primitive z^2-x^2=y^2 as well as introduce conception of the basic primitiveness of z^'2-x^'2=y^2 for z^'-x^'=1 and the dependant primitiveness of z^'2-x^'2=y^2 for co-prime x,y,z and z^'-x^'>1.
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Unlock a deep understanding of mathematics with our Module and Summary! Clear definitions, comprehensive discussions, relevant example problems, and step-by-step solutions will guide you through mathematical concepts effortlessly. Learn with a systematic approach and discover the magic in every step of your learning journey. Mathematics doesn't have to be complicated—let's make it simple and enjoyable!
S&S Game is an Mathematics Game for Junior High School Students in year 8. It created in order to help teachers do an interactive learning, especially in sequences and series topic for grade 8. In this platform, it's only as a file review and uploaded in pdf format, so the macro designed in this game was unabled to show. If you mind to use the game, it's free to ask the creator for the pptm format of the game, so you can use the game perfectly.
ملزمة الرياضيات للصف السادس العلمي الاحيائي - التطبيقيanasKhalaf4
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for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
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Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
1. Example A. The sequence 2, 6, 18, 54, … is a geometric
sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.
Since a1 = 2, set them in the general formula of the
geometric sequence an = a1r n – 1 , we get the specific formula
for this sequence an = 2(3n – 1).
Geometric Sequences
If a1, a2 , a3 , …an is a geometric sequence such that the
terms alternate between positive and negative signs,
then r is negative.
Example B. The sequence 2/3, –1, 3/2, –9/4, … is a geometric
sequence because
–1/(2/3) = (3/2) / (–1) = (–9/4) /(3/2) = … = –3/2 = r.
Since a1 = 2/3, the specific formula is
an = ( )n–12
3 2
–3
Use the general formula of geometric sequences
an = a1*rn–1 a to find the specific formula.
2. Example C. Given that a1, a2 , a3 , …is a geometric sequence
with r = –2 and a5 = 12,
a. find a1
By that the general geometric formula
an = a1r n – 1, we get
a5 = a1(–2)(5 – 1) = 12
a1(–2)4 = 12
16a1 = 12
a1 = 12/16 = ¾
3
4
an= (–2)n–1
Geometric Sequences
To use the geometric general formula to find the specific
formula, we need the first term a1 and the ratio r.
b. find the specific equation.
Set a1 = ¾ and r = –2 into the general formula an = a1rn – 1 ,
we get the specific formula of this sequence
3. set n = 9, we get
c. Find a9.
3
4
a9= (–2)9–1
a9 = (–2)8 = (256) = 192
3
4
Geometric Sequences
3
4
Since an= (–2)n–1,
3
4
Example D. Given that a1, a2 , a3 , …is a geometric sequence
with a3 = –2 and a6 = 54,
a. find r and a1
Given that the general geometric formula an = a1rn – 1,
we have
a3 = –2 = a1r3–1 and a6 = 54 = a1r6–1
–2 = a1r2 54 = a1r5
Divide these equations: 54
–2
=
a1r5
a1r2
4. 54
–2
=
a1r5
a1r2
–27
3 = 5–2
–27 = r3
–3 = r
Put r = –3 into the equation –2 = a1r2
Hence –2 = a1(–3)2
–2 = a19
–2/9 = a1
Geometric Sequences
b. Find the specific formula and a2
Use the general geometric formula an = a1rn – 1,
set a1 = –2/9, and r = –3 we have the specific formula
–2
9
an = (–3)n–1
–2
9
(–3) 2–1
To find a2, set n = 2, we get
–2
9
a2 =
3
2
3= (–3) =
5. Geometric Sequences
2
3
– 3
2
an= ( ) n–1
To find n, set an = =
2
3
– 3
2
( ) n – 1–81
16
– 3
2
= ( ) n – 1–243
32
Compare the denominators to see that 32 = 2n – 1.
Since 32 = 25 = 2n – 1
n – 1 = 5
n = 6
= a
1 – rn
1 – r
The Sum of the First n Terms of a Geometric Sequence
a + ar + ar2 + … +arn–1
Example E. Find the geometric sum :
2/3 + (–1) + 3/2 + … + (–81/16)
We have a = 2/3 and r = –3/2, and an = –81/16. We need the
number of terms. Put a and r in the general formula to get the
specific formula
6. Therefore there are 6 terms in the sum,
2/3 + (–1) + 3/2 + … + (–81/16)
S =
2
3
1 – (–3/2)6
1 – (–3/2)
=
2
3
1 – (729/64)
1 + (3/2)
=
2
3
–665/64
5/2
–133
48
Geometric Sequences
Set a = 2/3, r = –3/2 and n = 6 in the formula
1 – rn
1 – r
S = a
we get the sum S
=
7. Infinite Sums of Geometric Sequences
The Sum of Infinitely–Many Terms of a Geometric Sequence
Given a geometric sequence a, ar, ar2 … with| r | < 1
a rn = a + ar + ar2 + … = a
1 – rn=0
∞
then
google source
15 cm2
Assuming the ratio of 1.15
is the cross–sectional areas of
the successive chambers so
the areas of the chambers form
a geometric sequence,
starting with the first area of
15 cm2 with r = 1/1.15.
Hence the approximate total
area is the infinite sum:
15
1 – (1/1.15)n=0
∞
15 + 15(1/1.15) + 15(1/1.15)2 + 15(1/1.15)23 + ...
= 15(1/1.15)n =
8. Geometric Sequences
2. –2, 4, –8, 16,..1. 1, 3, 9, 27,..
4. 3/64, 9/32, 27/64, 81/128,..3. 1/90, 1/30, 1/10, 3/10,..
6. 2.3, 0.23, 0.023, 0.0023,..5. 4/3, – 2/3, 1/3, –1/6,..
8. a3 = –17,.., r = 1/2,7. a2 = 3/16,.., r = –2,
10. a5 = 4, r = –1/39. a4 = –2, r = 2/3
12. a3 = 125, a6 = –111. a4 = 0.02, a7 = 20
15. a2 = 0.3, a4 = 0.003
Exercise A. For each geometric sequence below
a. identify the first term a1 and the ratio r
b. find a specific formula for an and find a10
c. find the sum an
d. if –1 < r < 1, find the sum an. Use a calculator if needed.
n=1
20
n=1
∞
16. a4 = –0.21, a8 = – 0.000021
13. a4 = –5/2, a8 = –40 14. a3 = 3/4, a6 = –2/9
9. Geometric Sequences
2. –2 + 6 –18 + .. + 486
3. 6 – 3 + 3/2 – .. + 3/512
1. 3 + 6 + 12 + .. + 3072
4. 4/3 + 8/9 + 16/27 + 32/81
5. We deposit $1,000 at the beginning of each month at
1% monthly interest rate for 10 months. How much is there
in total right after the last or the 10th deposit?
6. Find a formula for the total right after the kth deposit.
B. For each sum below, find the specific formula of the
terms, write the sum in the notation, then find the sum.