Question

The seventh term of an Arithmetic progression is four times its second

term and twelfth term is 2 more than three times of its fourth term. Find

the progressi​

Answer 1

its common difference is d. A.T.Q. "The 7th term of an arithmetic progression is four times its second term and twelfth term is 2 more than three times of its fourth term. ... The n th term of the A.P a, a+d, a+2d

Answer 2

S O L U T I O N :

$\bigstar$ Firstly, we know that formula of the A.P;

$\boxed{\bf{a_n=a+(n-1)d}}}}}$

• a is the first term
• d is the common difference.
• n is the term of an A.P.

A/q

$\longrightarrow\sf{a_7=4a_2}\\\\\longrightarrow\sf{a+(7-1)d=4[a+(2-1)d]}\\\\\longrightarrow\sf{a+6d=4(a+d)}\\\\\longrightarrow\sf{a+6d=4a+4d}\\\\\longrightarrow\sf{a-4a=4d-6d}\\\\\longrightarrow\sf{\cancel{-}3a=\cancel{-}2d}\\\\\longrightarrow\sf{a=2d/3.................(1)}$

&

$\longrightarrow\sf{a_{12}=2+3a_4}\\\\\longrightarrow\sf{a+(12-1)d=2+3[a+(4-1)d]}\\\\\longrightarrow\sf{a+11d=2+3(a+3d)}\\\\\longrightarrow\sf{a+11d=2+3a+9d}\\\\\longrightarrow\sf{a-3a+11d-9d=2}\\\\\longrightarrow\sf{-2a+2d=2}\\\\\longrightarrow\sf{-2\bigg(\dfrac{2d}{3} \bigg)+2d=2\:\:[from(1)]}\\\\\longrightarrow\sf{\dfrac{-4d}{3} +2d=2}\\\\\longrightarrow\sf{-4d+6d=6}\\\\\longrightarrow\sf{2d=6}\\\\\longrightarrow\sf{d=\cancel{6/2}}\\\\\longrightarrow\bf{d=3}$

Putting the value of d in equation (1),we get;

$\longrightarrow\sf{a=\dfrac{2(3)}{3} }\\\\\\\longrightarrow\sf{a=\cancel{\dfrac{6}{3} }}\\\\\\\longrightarrow\bf{a=2}$

$\boxed{\bf{Arithmetic\:progression\::}}}}}$

$\bullet\:\sf{a=\boxed{\bf{2}}}}\\\\\bullet\sf{a+d=2+3=\boxed{\bf{5}}}}\\\\\bullet\sf{a+2d=2+2(3)=2+6=\boxed{\bf{8}}}}\\\\\bullet\sf{a+3d=2+3(3)=2+9=\boxed{\bf{11}}}}$