Question

Find out which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference. $p,p+90,p+180,p+270,.....$ where $p={\left(999\right)}^{999}$

Answer 1

Given sequence is,

$p,p+90,p+180,p+270,...\dots$

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

second term of this A.P is $a_2=p+90$

third term of this A.P is $a_3=p+180$

fourth term of this A.P is $a_4=p+270$

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

$d=a_2-a_1=(p+90)-\left(p\right)=90$

putting n=2 in above equation

$d=a_3-a_2=(p+180)-(p+90)=90$

putting n=3 in above equation

$d=a_4-a_3=(p+270)-(p+180)=90$

as we can see we get a common difference $d=90$ for this sequence

hence this sequence forms an A.P

with first term $a_1=p$ and

common difference $d=90$