In an arithmetic sequence, the sum of the 4th and 9th term is 89. The sum of the 2nd and 6th term is 54. What is the 3rd term?

1 Answer
Oct 27, 2015

Answer:

The third term is #20#

Explanation:

For an arithmetic sequence with an initial value of #a# and an increment of #b#:
#{: ("term",color(white)("XXX"),1st,2nd,3rd,4th,5th,6th,7th,8th,9th), ("value",color(white)("XXX"),a,a+1b,a+2b,a+3b,a+4b,a+5b,a+6b,a+7b,a+8b) :}#

Sum of #4th# and #9th# terms
#color(white)("XXX")(a+3b)+(a+8b)=89#
Simplified as
[1]#color(white)("XXX")2a+11b=89#

Sum of #2nd# and #6th# terms
#color(white)("XXX")(a+1b)+(a+5b)=54#
Simplified as
[2]#color(white)("XXX")2a+6b=54#

Subtracting [2] from [1]
[3]#color(white)("XXX")5b=35#

#rArr#[4#]color(white)("XXX")b=7#

Substituting #7# for #b# in [1]
[5]#color(white)("XXX")2a+11(7) =89#

#rArr#[6]#color(white)("XXX")a=6#

#3rd# term #=a+2b = 6+2(7) = 20#