# 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?

Oct 27, 2015

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 $4 t h$ and $9 t h$ terms
$\textcolor{w h i t e}{\text{XXX}} \left(a + 3 b\right) + \left(a + 8 b\right) = 89$
Simplified as
[1]$\textcolor{w h i t e}{\text{XXX}} 2 a + 11 b = 89$

Sum of $2 n d$ and $6 t h$ terms
$\textcolor{w h i t e}{\text{XXX}} \left(a + 1 b\right) + \left(a + 5 b\right) = 54$
Simplified as
[2]$\textcolor{w h i t e}{\text{XXX}} 2 a + 6 b = 54$

Subtracting [2] from [1]
[3]$\textcolor{w h i t e}{\text{XXX}} 5 b = 35$

$\Rightarrow$[4]color(white)("XXX")b=7

Substituting $7$ for $b$ in [1]
[5]$\textcolor{w h i t e}{\text{XXX}} 2 a + 11 \left(7\right) = 89$

$\Rightarrow$[6]$\textcolor{w h i t e}{\text{XXX}} a = 6$

$3 r d$ term $= a + 2 b = 6 + 2 \left(7\right) = 20$