Jeopardy Unit 7

Arithmetic & Geometric

100

What is the explicit formula of a geometric sequence?

u_n = ar^n-1

with first term a and ratio r.

200

Find the explicit formula for the nth term of an arithmetic sequence with first term 9 and second term 12.

u_n = 9 + (n-1)*3

300

Find the explicit formula for the nth term of a geometric sequence with third term 25 and ratio 5.

u_n = 1*5^n-1

400

An arithmetic sequence has second term 12 and third term 16. Give the explicit formula and find u_6.

u_n = 8 + (n-1)4

u₆ = 8 + 5*4 = 8 + 20 = 28.

500

The owners of a new coffee shop are trying to work out how their customers numbers will grow. At the opening of the cafe there are 20 people and the owners think that they will get 11 new customers each month.

How many customers will they have after one year?

Arithmetic sequence

u_n= 20 + (n-1)*11

u₁₂= 20 + 11*11 = 20 + 121 = 141 customers.

Linear & Quadratic

100

What is the next term in this sequence?

3, 8, 13, 18, ...

23

200

Find the first 4 terms of the following sequence:

u_n = 4n - 1 

u₁ = 4*1 - 1 = 3

u₂ = 4*2 - 1 = 7

u₃ = 4*3 - 1 = 11

u₄ = 4*4 - 1 = 15

300

Find the second difference of the sequence: 

6, 11, 19, 30, 44, 61

Difference diagram:

6, 11, 19, 30, 44, 61

First difference: 5, 8, 11, 14, 17

Second difference: 3, 3, 3, 3, thus second difference = 3

400

An elevator in 10 story building stops at each floor. It takes 6 seconds to travel to the first floor, 12 seconds to the second floor, 18 seconds to the third floor and 24 seconds to the fourth floor. 

How many seconds does it take to get to the 10th floor?

It's a linear sequence, with first difference 6.

Thus u_n = 6n.

u₁₀ = 6*10 = 60 seconds.

500

Find a general formula for the nth term of the following sequence, which has second difference 6.

6, 18, 36, 60

6/2 = 3.

Make a table with u_n, 3n² and u_n- 3n².

u_n - 3n² gives the row: 3 - 6 - 9 - 12

Thus u_n - 3n² = 3n. --> u_n = 3n² + 3n.

Simplifying

100

What are two forms of generalizing? 

1: The creation of more general problem in the hope of simplifying a problem. (Simplifying problems)

2: The creation of a conjecture based on a collection of pieces of evidence. (Making conjectures)

200

I want to solve 1054²-1052², but I can't use a calculator. I want to sipmlify the problem by generalizing it, using n. What value do I choose for n?

n = 1053.

300

What do I choose for n, if I want to solve the following problem by generalizing it?

(1001² - 499²) / 1500²

n=500

400

Solve 401² - 399² by generalizing the problem.

(Don't use a calculator)

Choose n=400

Solve by substituting n: (n+1)² - (n-1)²

Working out the brackets will give you 4n.

Thus 401² - 399² = 4n = 4*400 = 1600.

500

Solve the following problem by generalizing:

(601² - 299²) / 900²

(Don't use a calculator)

Choose n=300 and solve for n:

((2n+1)² - (n-1)²) / (3n)² = (4n²+4n+1-n²+2n-1) / 9n² = (3n²+6n) / 9n² = 1/3 + 2/(3n)

Thus: (601² - 299²) / 900² = 1/3 + 2/(3*300) = 1/3 + 2/900. (= 300/900 + 2/900 = 302/900)

Conjectures

100

What do we call a conjecture that has been proven?

A theorem.

200

Someone says:

“The sum of two consecutive integers is always odd.”

Is this conjecture true or false? 

True.

Two consecutive integers always consist of one even number and one odd number. Even + odd = odd.

300

Complete the conjecture:

The square of an even number is always ______.

The square of an even number is always even.

400

Make a conjecture about adding 2 even numbers together.

First try it out: 2+6=8, 4+10=14, 22+24=46, etcetera.

Then form your conjecture: 'Adding two even numbers together will always result in another even number.'

500

Make a general statement (conjecture) about adding three consecutive integers together. 

Try it out: 2+3+4=9, 4+5+6=15, 10+11+12=33, ..

General statement: 'adding three consecutive integers together will always be divisible by 3.' 

Or: 'adding three consecutive integers together willalways divisible by the middle integer'.

(Prove: n + (n+1) + (n+2) = 3n + 3 = 3(n+1). Since 3 and 'n+1' are multiples, it is always divisible by those).

Random

100

What is the next term in this sequence?

4, 7, 10, 13, ...

16

200

What is always constant in a linear sequence and what is always constant in a quadratic sequence?

Linear: the first difference

Quadratic: the seccond difference

300

What is always constant in an arithmetic sequence and what is always constant in a geometric sequence?

Arithmetic: the difference between consecutive terms is always constant.

Geometric: the ratio between consecutive terms is always constant.

400

What kind of sequence is this and why?

3, 9, 27, 81, 243

A geometric sequence, since the ratio between consecutive terms is constant. The ratio is 3.

500

What can you say about the product of two consecutive integers?

Give examples and make a general statement.

2*3=6, 4*5=20, 7*8=56 10*11=110.

The product of two consecutive integers is always even.