Question

NO LINKS!!

Write the first 6 terms of the sequence beginning with the term a1. Then calculate the first and second differences of the sequence.
a1= 4
a_n= a_(n-1) + 5

a_n= ______, ______, ______, ______, ______, ______
first differences: _____, ______, _____, _____, _____
second differences: _____, ______, ______, ______

State whether the sequence has a linear model, a quadratic model or neither.

Answer 1

Explanation:

The first six terms of the given sequence are:

a_1 = 4

a_2 = a_1 + 5 = 4 + 5 = 9

a_3 = a_2 + 5 = 9 + 5 = 14

a_4 = a_3 + 5 = 14 + 5 = 19

a_5 = a_4 + 5 = 19 + 5 = 24

a_6 = a_5 + 5 = 24 + 5 = 29

The first differences of the sequence are:

9 - 4 = 5

14 - 9 = 5

19 - 14 = 5

24 - 19 = 5

29 - 24 = 5

The second differences of the sequence are:

5 - 5 = 0

5 - 5 = 0

5 - 5 = 0

5 - 5 = 0

Since the second difference of the sequence are all equal to 0, the sequence has a linear model. The model of the sequence can be written as:

a_n = 5n + (-1)

This means that each term in the sequence is 5 more than the previous term, starting with the term a_1 = 4.

Answer 2

First 6 terms: 4, 9, 14, 19, 24, 29, ...

First differences: +5, +5, +5, +5, +5

Second differences: +0, +0, +0, +0

Linear model

Explanation:

A recursive formula for an arithmetic sequence allows you to find the nth term of the sequence provided you know the value of the previous term in the sequence.

Given recursive formula:

`\begin{cases}a_1= 4\\a_n= a_(n-1) + 5\end{cases}`

Calculate the first 6 terms of the sequence:

`a_1=4`

`a_2=a_(2-1)+5=a_1+5=4+5=9`

`a_3=a_(3-1)+5=a_2+5=9+5=14`

`a_4=a_(4-1)+5=a_3+5=14+5=19`

`a_5=a_(5-1)+5=a_4+5=19+5=24`

`a_6=a_(6-1)+5=a_5+5=24+5=29`

Therefore, the first 6 terms of the sequence are:

• 4, 9, 14, 19, 24, 29, ...

Calculate the first differences between the terms:

`4\underset{+5}{\longrightarrow}9\underset{+5}{\longrightarrow}14\underset{+5}{\longrightarrow}19\underset{+5}{\longrightarrow}24\underset{+5}{\longrightarrow}29`

Calculate the second differences (i.e. the differences between the first differences):

`5\underset{+0}{\longrightarrow}5\underset{+0}{\longrightarrow}5\underset{+0}{\longrightarrow}5\underset{+0}{\longrightarrow}5`

As the first differences are constant and the second differences are zero, the sequence is a linear model.