The third term of an arithmetic sequence is 14, and the ninth term is -1. How do you find the first four terms of the sequence?

Question

The third term of an arithmetic sequence is 14, and the ninth term is -1.  How do you find the first four terms of the sequence?

Cora Alexander · Accepted Answer

#19" "16.5" "14" "11.5" " ....#

You can write an equation (or formula) for each term in an AP.

#T_n = a + d(n-1)#

"The third term is 14" can be written with the formula as:

#color(blue)(T_3 = a +2d = 14)" "rarr(n-1) = 3-1 = 2#

"The ninth term is -1 " can be written as:

#color(blue)(T_9= a +8d= -1)" "rarr(n-1) = 9-1 =8#

Now there are two equations with two unknowns. Solve them simultaneously (as a system).

#color(blue)(T_9= a +8d= -1#............................A #color(blue)(T_3 = a +2d = +14)# ........................B

A-B: #" "6d = -15# #" "d = -15/6 = -5/2" "larr#this is the value for d

#color(blue)(T_3 = a +2((-5)/2) = +14)#

#a -5 = 14# #a = 19" "larr# this is the value for a

This sequence is given by #color(red)(T_n = 19 -5/2d)#

#T_1 = a = 19# For #T_2: n = 2 rarr 19-5/2(1) = 16.5# #T_3: 16.5-5/2 = 14# #T_4: 14-5/2 = 11.5#

Each term is 2.5 less than the previous term:

#19" "16.5" "14" "11.5" "9 ....#