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

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;

Recursive formula:

Question 2

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

This implies that,

This implies that,

The explicit rule:

The recursive rule is

where,

step-by-step explanation:

given that there are two triangles j'k'l' and jkl>

the former is got by dilating the second one.

when drawn on a grid, (graph paper) coordinates of vertices of first triangle is

(6,12)j, (9,6)k and (12,9)l

side jk has length = distance between j and k

=

side j'k'=distance between j' and k'

=

ie j'k'l' has sides 1/3 of that of corresponding sides of jkl.

hence scale factor of 1/3 is used to get j'k'l'from jkl