# 2n2>(n+1)2,n≥32n2>(n+1)2,n≥3

**THE LOTTERY**

**What does she know about mass psychology?**

I believe that the “she” in your question is referring to the author of “The Lottery,” Shirley Jackson. I believe that Jackson understands mass psychology quite well. For example, she clearly…

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**SCIENCE**

**The lowest pressure achieved in a laboratory is about 1.0 X 10^-15 mm Hg. How many molecules of…**

In this question, we have to use the ideal gas law, which can be written as: PV = nRT where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant and T is…

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**THE ADVENTURES OF HUCKLEBERRY FINN**

**What story do the travelers on the raft hear from the innocent young man they encounter, and how…**

In chapter 24, the innocent young man on the raft tells the King about a deceased man from the area who is named Peter Wilks. Before he died, this wealthy man sent for Harvey and William Wilks, his…

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**THE BET**

**In “The Bet,” why does the narrator call the bet “wild” and “senseless”?**

A lively discussion arose. The banker, who was younger and more nervous in those days, was suddenly carried away by excitement; he struck the table with his fist and shouted at the young man:…

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**PRIDE AND PREJUDICE**

**What is Mr. Collins’s view on his position in society in Pride and Prejudice?**

Mr. Collins seems to believe that, with Lady Catherine de Bourgh as his patroness, his position in society is somewhat higher than it actually is. Though he has some education, he is a silly man…

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**HISTORY**

**Who is Saint Augustine? What are some important things about him, and what are some important…**

Saint Augustine of Hippo lived from 354 CE to 430 CE in modern-day Algeria. He developed the concept of the City of God- a Catholic city which exists in spirit but not necessarily on Earth. Though…

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**SCIENCE**

**What is the difference between an endergonic reaction and an exergonic reaction?**

Some reactions need energy to take place, while others release energy when they occur. Depending on whether the energy is needed by the reaction or released by the reaction, a given reaction can be…

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**THE HOUND OF THE BASKERVILLES**

**What literary techniques does Doyle use in The Hound of the Baskervilles?**

Some of the techniques used by Sir Arthur Conan Doyle in The Hound of the Baskervilles include first person narration, foreshadowing, Gothic setting, red herrings, irony, symbolism, and foils. As…

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**MACBETH**

**How is the relationship between Macbeth and Lady Macbeth being demonstrated?**

The relationship between Macbeth and Lady Macbeth is first demonstrated by Macbeth’s letter to her. In it, he calls her his “dearest partner of greatness” and says that he wanted to write to her,…

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**HISTORY**

**How are WWI and the Spanish-American War alike and different?**

Differences The Spanish-American War was a war between two countries while World War 1 was a war between many countries. In WW1, the countries included Germany, Britain, Russia, and Austria-Hungary…

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**SCIENCE**

**What are the six characteristics animals share?**

According to the biologists Gerald and Teresa Audesirk, all members of the animal kingdom have six characteristics in common. They are as follows: All animals are made up of cells that do not have…

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**MATH**

**a0=−3,a2=−5,a6=−57a0=-3,a2=-5,a6=-57 Find a quadratic model for the sequence with the indicated…**

You need to remember what a quadratic model is, such that: an=f(n)=a⋅n2+b⋅n+can=f(n)=a⋅n2+b⋅n+c The problem provides the following information, such that:

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**MATH**

**a1=0,a2=8,a4=30a1=0,a2=8,a4=30 Find a quadratic model for the sequence with the indicated terms.**

You need to remember what a quadratic model is, such that: an=f(n)=a⋅n2+b⋅n+can=f(n)=a⋅n2+b⋅n+c The problem provides the following information, such that:

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**MATH**

**a0=3,a2=0,a6=36a0=3,a2=0,a6=36 Find a quadratic model for the sequence with the indicated terms.**

You need to remember what a quadratic model is, such that: an=f(n)=a⋅n2+b⋅n+can=f(n)=a⋅n2+b⋅n+c The problem provides the following information, such that:

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**MATH**

**a0=−3,a2=1,a4=9a0=-3,a2=1,a4=9 Find a quadratic model for the sequence with the indicated terms.**

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**MATH**

**a0=7,a1=6,a3=10a0=7,a1=6,a3=10 Find a quadratic model for the sequence with the indicated terms.**

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**MATH**

**a0=3,a1=3,a4=15a0=3,a1=3,a4=15 Find a quadratic model for the sequence with the indicated terms.**

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**MATH**

The given sequence is: −1,8,23,44,71,104-1,8,23,44,71,104 To determine if it is a linear sequence, take the difference between consecutive terms. −1,8,23,44,71,104-1,8,23,44,71,104 ⋁⋁ ⋁⋁ ⋁⋁ ⋁⋁…

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**MATH**

**−2,1,6,13,22,33…..-2,1,6,13,22,33….. Decide whether the sequence can be represented perfectly by a…**

Firstly we need to determine whether the series is linear or quadratic. A linear sequence is a sequence of numbers in which there is a first difference between any consecutive terms is constant….

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**MATH**

**0,6,16,30,48,70…0,6,16,30,48,70… Decide whether the sequence can be represented perfectly by a…**

Firstly we need to determine whether the series is linear or quadratic. A linear sequence is a sequence of numbers in which there is a first difference between any consecutive terms is constant….

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**MATH**

Firstly we need to determine whether the series is linear or quadratic. A linear sequence is a sequence of numbers in which there is a first difference between any consecutive terms is constant….

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**MATH**

**2,9,16,23,30,37…2,9,16,23,30,37… Decide whether the sequence can be represented perfectly by a…**

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**MATH**

**5,13,21,29,37,45…5,13,21,29,37,45… Decide whether the sequence can be represented perfectly by a…**

The given sequence is: 5,13,21,29,37,455,13,21,29,37,45 To determine if it is a linear sequence, take the difference between the consecutive terms. 5,13,21,29,37,455,13,21,29,37,45 ⋁⋁ ⋁⋁ ⋁⋁ ⋁⋁…

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**MATH**

**∑j=110(3−12j+12j2)∑j=110(3-12j+12j2) Find the sum using formulas for the sums of powers of…**

You need to evaluate the given sum using formulas for the sums of powers of integers, such that:

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**MATH**

**∑i=16(6i−8i3)∑i=16(6i-8i3) Find the sum using formulas for the sums of powers of integers.**

Given: ∑i=16(6i−8i3)∑i=16(6i-8i3) =∑i=166i−∑i=168i3=∑i=166i-∑i=168i3 =6∑i=16i−8∑i=16i3=6∑i=16i-8∑i=16i3Use the Sums of Powers of Integers to find the sums. 1+2+3+4+…+n=n(n+1)21+2+3+4+…+n=n(n+1)2…

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**MATH**

**∑n=120(n3−n)∑n=120(n3-n) Find the sum using formulas for the sums of powers of integers.**

You need to evaluate the given sum using formulas for the sums of powers of integers, such that: σ20n=1(n3−n)=σ20n=1n3−σ20n=1nσn=120(n3-n)=σn=120n3-σn=120n

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**MATH**

**∑n=16(n2−n)∑n=16(n2-n) Find the sum using formulas for the sums of powers of integers.**

∑n=16(n2−n)∑n=16(n2-n) The formulas using the sums of the powers of integers are ∑n=1nn2=n(n+1)(2n+1)6∑n=1nn2=n(n+1)(2n+1)6 and ∑n=1nn=n(n+1)2∑n=1nn=n(n+1)2

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**MATH**

**∑n=18n5∑n=18n5 Find the sum using formulas for the sums of powers of integers.**

∑n=18n5∑n=18n5 The formula for the sum using the sums of the powers of integers is, 15+25+35+……….n5=n2(n+1)2(2n2+2n−1)1215+25+35+……….n5=n2(n+1)2(2n2+2n-1)12 ∴∑n=18n5=82(8+1)2(2⋅82+2⋅8−1)12∴∑n=18n5=82(8+1)2(2⋅82+2⋅8-1)12…

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**MATH**

**∑n=15n4∑n=15n4 Find the sum using formulas for the sums of powers of integers.**

∑n=15n4∑n=15n4 The formula for the sum using the sums of the powers of integers is, 14+24+34+………..+n4=n(n+1)(2n+1)(3n2+3n−1)3014+24+34+………..+n4=n(n+1)(2n+1)(3n2+3n-1)30

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**MATH**

**∑n=110n3∑n=110n3 Find the sum using formulas for the sums of powers of integers.**

∑n=110n3∑n=110n3 The formula for the sum using the sums of powers of integers is ∑n=1nn3=n2(n+1)24∑n=1nn3=n2(n+1)24 ∴∑n=110n3=102(10+1)24∴∑n=110n3=102(10+1)24 =100⋅1124=100⋅1124 =25⋅121=25⋅121 =3025=3025 So the…

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**MATH**

**∑n=16n2∑n=16n2 Find the sum using formulas for the sums of powers of integers.**

∑n=16n2∑n=16n2 The formula for the sum using the sums of the powers of integers is ∑n=1nn2=n(n+1)(2n+1)6∑n=1nn2=n(n+1)(2n+1)6 ∴∑n=16n2=6(6+1)(2⋅6+1)6∴∑n=16n2=6(6+1)(2⋅6+1)6 =6⋅7⋅136=6⋅7⋅136 =7⋅13=7⋅13 =91=91So the sum…

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**MATH**

**∑n=130n∑n=130n Find the sum using formulas for the sums of powers of integers.**

∑n=130n∑n=130n The formula for the sum using the sums of powers of integers is ∑n=1nn=n(n+1)2∑n=1nn=n(n+1)2 ∴∑n=130=30(30+1)2∴∑n=130=30(30+1)2 =30⋅312=30⋅312 =15⋅31=15⋅31=465=465 So the sum is 465.

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**MATH**

**∑n=115n∑n=115n Find the sum using formulas for the sums of powers of integers.**

∑n=115n∑n=115n The formula for the sum using the sums for the powers of power of integers is ∑n=1nn=n(n+1)2∑n=1nn=n(n+1)2 ∴∑n=115n=15(15+1)2∴∑n=115n=15(15+1)2 =15⋅162=15⋅162 =15⋅8=15⋅8 =120=120 So the sum is 120.

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**MATH**

**12⋅3,13⋅4,14⋅5,15⋅6….1(n+1)(n+2)…12⋅3,13⋅4,14⋅5,15⋅6….1(n+1)(n+2)… Use mathematical induction…**

12⋅3,13⋅4,14⋅5,15⋅6,……….1(n+1)(n+2)12⋅3,13⋅4,14⋅5,15⋅6,……….1(n+1)(n+2) Let’s write down the first few sums of the sequence, S1=12⋅3=16=12(1+2)S1=12⋅3=16=12(1+2)…

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**MATH**

**14,112,124,140,…12n(n+1)14,112,124,140,…12n(n+1) Use mathematical induction to find a formula for…**

14,112,124,140,…….12n(n+1)14,112,124,140,…….12n(n+1) Let’s write down the first few sums of the sequence, S1=14S1=14 S2=14+112=3+112=412=13=22(2+1)S2=14+112=3+112=412=13=22(2+1)…

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**MATH**

**3,−92,274,−8183,-92,274,-818 Use mathematical induction to find a formula for the sum of the first…**

You need to notice that the sequence can be rewritten such that: 3120,(−1)1⋅3221,(−1)2⋅3322,…,(−1)n⋅3n+12n3120,(-1)1⋅3221,(-1)2⋅3322,…,(-1)n⋅3n+12n Notice that the sequence is a geometric sequence,…

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**MATH**

**1,5,9,13…1,5,9,13… Use mathematical induction to find a formula for the sum of the first n…**

1,5,9,13,……1,5,9,13,…… Now let’s denote the first term of the above sequence by a1a1 and k’th term by akak and let’s write down the first few sums of the sequence. S1=1S1=1…

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**MATH**

**2n2>(n+1)2,n≥32n2>(n+1)2,n≥3 Use mathematical induction to prove the inequality for the…**

You need to use mathematical induction to prove the formula for every positive integer n, hence, you need to perform the two steps of the method, such that: Step 1: Basis: Show that the statement…

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**MATH**

**(1+a)n≥na,n≥1anda>0(1+a)n≥na,n≥1anda>0 Use mathematical induction to prove the…**

You need to use mathematical induction to prove the inequality, hence, you need to perform the following two steps, such that: Step 1: Basis: Prove that the statement holds for n = 1

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**MATH**

**(xy)n+1<(xy)n,n≥1and0<x<y(xy)n+1<(xy)n,n≥1and0<x<y Use mathematical induction to prove…**

You need to use mathematical induction to prove the inequality, hence, you need to perform the following two steps, such that: Step 1: Basis: Prove that the statement holds for n = 1 If x<y then…

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**MATH**

**12–√+13–√+14–√+…1n−−√>n−−√,n≥212+13+14+…1n>n,n≥2 Use mathematical…**

To prove 11–√+12–√+13–√+14–√+……..1n−−√>n−−√11+12+13+14+……..1n>n n≥2≥2 For n=2, 11–√+12–√≈1.70711+12≈1.707 2–√≈1.4142≈1.414 ∴11–√+12–√>2–√∴11+12>2 Let’s assume…

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**MATH**

**n!>2nn≥4n!>2nn≥4 Use mathematical induction to prove the inequality for the indicated…**

You need to use mathematical induction to prove the formula for every positive integer n, hence, you need to perform the two steps of the method, such that: Step 1: Basis: Show that the statement…

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**MATH**

**(1+11)(1+12)(1+13)…(1+1n)=n+1(1+11)(1+12)(1+13)…(1+1n)=n+1 Use mathematical induction to prove the…**

You need to use mathematical induction to prove the formula for every positive integer n, hence, you need to perform the two steps of the method, such that: Step 1: Basis: Show that the statement…

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**MATH**

**12+32+52+…(2n−1)2=n(2n−1)2n+1312+32+52+…(2n-1)2=n(2n-1)2n+13 Use mathematical induction to prove…**

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**MATH**

**13+23+33+43+n3=n2(n+1)2413+23+33+43+n3=n2(n+1)24 Use mathematical induction to prove the…**

You need to use mathematical induction to prove the formula for every positive integer n, hence, you need to perform the two steps of the method, such that: Step 1: Basis: Show that the…

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**MATH**

**1+2+3+4+5…+n=n(n+1)21+2+3+4+5…+n=n(n+1)2 Use mathematical induction to prove the formula…**

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**MATH**

**2(1+3+32+33+…3n−1)=3n−12(1+3+32+33+…3n-1)=3n-1 Use mathematical induction to prove the…**

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**MATH**

**1+2+22+23+…2n−1=2n−11+2+22+23+…2n-1=2n-1 Use mathematical induction to prove the formula…**

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**MATH**

**1+4+7+10+…(3n−2)=n2(3n−1)1+4+7+10+…(3n-2)=n2(3n-1) Use mathematical induction to prove the…**

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**MATH**

**2+7+12+17+….(5n−3)=(n2)(5n−1)2+7+12+17+….(5n-3)=(n2)(5n-1) Use mathematical induction to prove the…**

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