`2, 9, 16, 23, 30, 37...` Decide whether the sequence can be represented perfectly by a linear or a quadratic model. If so, then find the model.
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. However, a quadratic sequence is a sequence of numbers in which there is a second difference between any consecutive terms is constant.
Now let's determine if the above sequence is linear or quadratic.
Lets begin by finding the first difference:
`T_2 - T_1 = 9 - 2 = 7`
`T_3 - T_2 = 16 - 9 = 7`
`T_4 - T_3 = 23 - 16 = 7`
From above we can see we have a constant number for the first difference, hence our sequence is linear.
Now let's determine the model of this sequence. The equation of a linear sequence is as follows:
`T_n = a + d(n-1)`
T_n = Value of the term in sequence
a = first number of sequence
d = common difference (first difference)
n = term number
Now let's substitute values into the above equation:
`T_n = 2 + 7(n-1)`
`T_n = 2 + 7n - 7`
The model is simplified to:
`T_n = 7n -5`
Now let's double check our model using terms 1, 3 and 6:
`T_1 = 7(1) - 5 =2`
`T_3 = 7(3) - 5 = 16`
`T_7 = 7(6) - 5 = 37`
The sequence is linear.
`T_n = 7n - 5`