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

The third term is 2020

Explanation:

For an arithmetic sequence with an initial value of aa and an increment of bb:
{: ("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