a Decide if the difference between any two consecutive terms is the same throughout the sequence.
B
b Decide if the difference between any two consecutive terms is the same throughout the sequence.
C
c Decide if the difference between any two consecutive terms is the same throughout the sequence.
D
d Decide if the difference between any two consecutive terms is the same throughout the sequence.
E
e Decide if the difference between any two consecutive terms is the same throughout the sequence.
A
a Yes.
B
b No.
C
c No.
D
d No.
E
e Yes.
Practice makes perfect
a We are given five number sequences and we need to decide whether or not they are arithmetic sequences. To do so, we will see if the difference between any two consecutive terms is always the same. Let's see the difference between consecutive terms in the first sequence.
f(n)-f(n-1)
=
d
f(2)-f(1)
7-6
1
f(3)-f(2)
8-7
1
f(4)-f(3)
9-8
1
f(5)-f(4)
10-9
1
As we can see, the common difference d between two consecutive terms is always 1. Therefore, it is an arithmetic sequence.
b Now, let's see the difference between consecutive terms in the second sequence.
f(n)-f(n-1)
=
d
f(2)-f(1)
10-5
5
f(3)-f(2)
20-10
10
This time, we can stop before checking all of the differences. The difference between two consecutive terms being different even once is enough to show that it's not always the same. Therefore, 5,10,20,35,55,... is not an arithmetic sequence.
c Again, let's find the difference between consecutive terms, this time in the third sequence.
f(n)-f(n-1)
=
d
f(2)-f(1)
-1-0
-1
f(3)-f(2)
1-(-1)
2
Just as in Part B, we can stop after just two calculations. We already know that the difference between consecutive terms is not always the same. Therefore, it is not an arithmetic sequence.
d Yet again, let's calculate the difference between consecutive terms in the given sequence.
f(n)-f(n-1)
=
d
f(2)-f(1)
16-1
15
f(3)-f(2)
81-16
65
Once again, the difference between consecutive terms is not always the same and, therefore, the given sequence is not arithmetic.
e One last time, we can see the difference between consecutive terms.
f(n)-f(n-1)
=
d
f(2)-f(1)
- 4-(- 2)
-2
f(3)-f(2)
- 6-(- 4)
-2
f(4)-f(3)
- 8-(- 6)
-2
f(5)-f(4)
- 10-(- 8)
-2
This time, the common difference d between consecutive terms is always - 2. Therefore, this sequence is arithmetic.
