Question

Find out which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference. $1^2,3^2,5^2,7^2,...$

Answer 1

Given sequence is,

$1^2,3^2,5^2,7^2...\dots$

first term of this A.P is $a_1=1^2=1$

second term of this A.P is $a_2=3^2=9$

third term of this A.P is $a_3=5^2=25$

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

$d=a_2-a_1=9-1=8$

putting n=2 in above equation

$d=a_3-a_2=25-9=16$

as we can see we get two values of d, but in an A.P their must be a single value of d which means for this sequence we can't define a common difference hence this sequence not form an A.P