Question

the sum of the second and fifth terms of an ap is 22 and the sum of its fourth in the 9th term is 46 find the AP

Answer 1

Dear sttudent,
Let the terms of the AP are $a,a+d,a+2d,...$
Since, the sum of the second and fifth terms of the AP is 22.
Hence,
$$\begin{gathered} a+d+a+4d=22 \\ 2a+5d=22...\left(1\right) \end{gathered}$$
Since, the sum of its fourth and the 9th term is 46.
Hence,
$$\begin{gathered} a+3d+a+8d=46 \\ 2a+11d=46...\left(2\right) \end{gathered}$$
put $2a=22-5d$ form first equtaion in the second equtaion,
$$\begin{gathered} 22-5d+11d=46 \\ 6d=46-22 \\ d=\frac{24}{6}=4 \end{gathered}$$
so, $$\begin{gathered} 2a=22-5d=22-5\times 4=22-20=2 \\ 2a=2 \\ a=1 \end{gathered}$$
Therefore the AP is, $1,5,9,13,...$
Regards