Question

Find the first term and the common difference of the arithmetic sequence whose 11th term is 26 and 13th term is 32 .First term is Common difference is

Answer 1

We have a sequence which have the 11th term equal to a11 = 26 and the 13th term is a13 = 32.

If the difference is d, then, we can write:

$\begin{gathered} a_{12}=a_{11}+d \\ a_{13}=a_{12}+d \\ \Rightarrow a_{13}=a_{11}+d+d=a_{11}+2d \\ a_{13}-a_{11}=2d \\ d=\frac{a_{13}-a_{11}}{2} \\ d=\frac{32-26}{2} \\ d=\frac{6}{2} \\ d=3 \end{gathered}$

The common difference has to be d = 3.

We can use this to find the first term as:

$\begin{gathered} a_2=a_1+d \\ a_3=a_2+d=a_1+2d \\ \Rightarrow a_{11}=a_1+(11-1)d=a_1+10d \\ a_1=a_{11}-10d \\ a_1=26-10*3 \\ a_1=26-30 \\ a_1=-4 \end{gathered}$

Answer:

First term is -4.

Common difference is 3.