Question

Consider the following number pattern and answer the questions that follow:

12; 21; 30; 39

1) Fill in term number 7
2) Determine the general rule
3) Determine the 56th term

Answer 1

a₇ = 66 ,
`a_(n)` = 9n + 3 , a₅₆ = 507

Explanation:

(1)

there is a difference of + 9 between consecutive terms, that is

21 - 12 = 30 - 21 = 39 - 30 = 9

add 9 to each term to find the next term in the pattern

a₄ = 39

a₅ = a₄ + 9 = 39 + 9 = 48

a₆ = a₅ + 9 = 48 + 9 = 57

a₇ = a₆ + 9 = 57 + 9 = 66

that is the 7th term a₇ = 66

(2)

since there is a common difference between consecutive terms, then this is an arithmetic sequence with nth term

`a_(n)` = a₁ + d(n - 1)

where a₁ is the first term and d the common difference

here a₁ = 12 and d = 9 , then

`a_(n)` = 12 + 9(n - 1) = 12 + 9n - 9 = 9n + 3

general rule is
`a_(n)` = 9n + 3

(3)

use the general rule with n = 56 to obtain 56th term

a₅₆ = 9(56) + 3 = 504 + 3 = 507