Question

Can someone explain this pls

Answer 1

3) 11/12, 19/12, 9/4

4)

Part a)

Term 1:

-31

Term 2:

-38

Term 3:

-45

Term 4:

-52

Term 5:

-59

Part b) -24 - 7n

5) 5461

Explanation:

Knowledge Needed for Q3

An arithmetic sequence has 3 things you need to understand.

`a_n` : Commonly used for defining a sequence, keep it as a variable.

n: The number of the term in the sequence. For example, If you are trying to find the 3rd term of the sequence, n is 3. Replace n with the number for the sequence you are trying to find.

1/4: The base, or what the sequence starts with. Base is defined as
`a_1`, or the first term.

2/3n: This is addition/subtraction factor, or how much you add or subtract when ever the value of the term goes up by one.

For example, if they sequence adds 2 every time and the base is 4, the next 3 numbers are:

Our Formula: 4+2n

Replace n with the number of the term you want to find to find the value of that term.

4 + 2(1) = 6

4 + 2(2) = 8

4 + 2(3) = 10

Examples with terms:

Term 1: 6

Term 2: 8

Term 3: 10

The base is 4, the starting number.

The change factor is 2, as we increase by 2 every time the number of the term increases by 1.

Question 3

Question 3's Formula:

1/4 + 2/3n

To find the 1st term, replace n with 1 and solve/simplify like you would with an algebraic expression/ algebraic equation.

To find the 2nd term, replace n with 2 and solve/simplify like you would with an algebraic expression/ algebraic equation.

To find the 3rd term, replace n with 3 and solve/simplify like you would with an algebraic expression/ algebraic equation.

1st Term

1/4 + 2/3(1)

1/4 + 2/3

11/12

2nd Term

1/4 + 2/3(2)

1/4 + 4/3

19/12

3rd Term

1/4 + 2/3(3)

1/4 + 6/3

1/4 + 2

9/4

Knowledge Needed for Q4

Common Difference (d): What I described as the subtraction factor.

Explicit Formula: Simplified Formula to Find the Value from a Given Number of a Term

Q4

If
`a_3_6` is -276, it means 36 is the term number. -7 is the common difference. This means you subtract by 7 each time the term increases.

To find
`a_1`, lets use subtract the term number we are trying to find (1) from the term number we have (36) Then, we multiply the difference of 36 and 1 by 7 and add that to -276.

36 - 1 = 35

35 * 7 = 245

-276 + 245 = -31

Term 1:

`a_1 = -31`

To find the other 4 terms, subtract by 7.

Term 2:

-38

Term 3:

-45

Term 4:

-52

Term 5:

-59

To make the equation, we know the first term is -31. Every time we increase by 1 term, subtract by 7.

IMPORTANT

-31 - 7n

Here's the confusing part about sequences. Why did I write n-1?

Lets see. We know term 1 is -31. Plug 1 into 1 to try to solve.

-31 - 7(1)

-31 - 7

-38

Term 1 isn't -38! It's -31! This is because in terms of
`a_n`, n=1 is the first term and the base, but in an equation, n=0 is the first. To solve this problem, offset the n value by 1.

1 - ? = 0

1

So, our new equation is:

-31 - 7(n-1)

Simplify this equation using the distributive property for the explicit formula.

-31 - 7n + 7

-24 - 7n

Knowledge Needed For Q5

This way of expressing a sequence is called sigma notation (summation). This is used later on in Calculus too with limits. An example too help you understand.'

Σ

Number on top of the sigma: 4

(The index of the summation) Equation on the bottom of the sigma: k = 1

Expression to the right of the sigma:
`k^2`

Plug in 1 into the expression:

`(1)^2`

1 x 1 = 1

Then, plug in numbers until you hit the number on top of the sigma, 4. This includes 4.

These numbers are 2, 3, and 4.

Plug in:

`(2)^2 = 4\\(3)^2 = 9\\(4)^2 = 16`

Add all of the numbers (1, 4, 9 , 16).

1 + 4 + 9 + 16 = 30

Q5

Plug 1 into the expression (replace it with k, the variable.)

`4^(^1^-^1^)`

Do this with all numbers till 7 including 7.

`4^(^2^-^1^)`

`4^(^3^-^1^)`

`4^(^4^-^1^)`

`4^(^5^-^1^)`

`4^(^6^-^1^)`

`4^(^7^-^1^)`

Simplify.

4^0

4^1

4^2

4^3

4^4

4^5

4^6

Evaluate.

1

4

16

64

256

1024

4096

Add:

5461