Question

Find out which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference. $a+b,(a+1)+b,(a+1)+(b+1),(a+2)+(b+1),(a+2)+(b+2),....$

Answer 1

Given sequence is,

$a+b,(a+1)+b,(a+1)+(b+1),(a+2)+(b+1),(a+2)+(b+2)...\dots$

first term of this A.P is $a_1=a+b$

second term of this A.P is $a_2=(a+1)+b$

third term of this A.P is $a_3=(a+1)(b+1)$

fourth term of this A.P is $a_4=(a+2)+(b+1)$

fifth term of this A.P is $a_5=(a+2)+(b+2)$

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

$d=a_2-a_1=(a+1+b)-(a+b)=1$

putting n=2 in above equation

$d=a_3-a_2=(a+1+b+1)-(a+1+b)=1$

putting n=3 in above equation

$d=a_4-a_3=(a+2+b+1)-(a+1+b+1)=1$

putting n=4 in above equation

$d=a_5-a_4=(a+2+b+2)-(a+2+b+1)=1$

as we can see we get a common difference $d=1$ for this sequence

hence this sequence forms an A.P

with first term $a_1=a+b$ and

common difference $d=1$