Question

Write a rule to find the nth term for an arithmetic sequence given the following:

a3 = 14
a12 = 59

Recursive rule: ?
Explicit rule: ?

Write an explicit and recursive rule for a sequence given the following:

a4 = 2
r = 1/3

Recursive rule: ?
Explicit rule: ?

Answer 1

I'm having trouble with this type of math to so your not alone

Explanation:

Answer 2

See explanation

EXPLANATION

Question 1:

The third term of the arithmetic sequence is :

14=a+2d...(1)

The twelveth term is

59=a+11d...(2)

Subtract equation (1) from (2)

45=9d

This implies that

d=5

a=14-2(5)=4

The explicit rule is;

`a_(n)=4 + 5(n - 1)`

`a_(n)=4 + 5n -5`

`a_(n) = 5n -1`

Recursive formula:

`a_(n)=a_(n - 1) + 5`

Question 2

The geometric sequence has the fourth term to be 2 and the common ratio to be r=⅓

This implies that,

`a {( (1)/(3) })^(3) = 2`

This implies that,

`(a)/(27) = 2`

`a = 54`

The explicit rule:

`a_n=54 {( (1)/(3) })^(n - 1)`

The recursive rule is

`a_n=( (1)/(3) )a_(n-1)`

where,

`a_1 = 54`