Question

Find out which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference. $12,2,-8,-18,......$

Answer 1

Given sequence is,

$12,2,-8,-18,...\dots$

first term of this A.P is $a_1=12$

second term of this A.P is $a_2=2$

third term of this A.P is $a_3=-8$

fourth term of this A.P is $a_4=-18$

the condition for an sequence to be an A.P is their must be a common difference $(i.e.,d=a_{n+1}-a_n)$

putting n=1 in above equation

$d=a_2-a_1=-2-12=-10$

putting n=2 in above equation

$d=a_3-a_2=-8-2=-10$

putting n=3 in above equation

$d=a_4-a_3=-18-(-8)=-10$

as we can see we get a common difference $d=-10$ for this sequence

hence this sequence forms an A.P

with first term $a_1=12$ and

common difference $d=-10$