Question

Show that a sequence is an A.P. if its nth term is a linear expression in n and in such a case the common difference is equal to the coefficient of n

Answer 1

a/c let Tn =an+c is a linear equation where a is coefficient of n and c is constant
first of all
we put n=1
T1=a+c
put n=2
T2=2a+c
put n=3
T3=3a+c
put n=4
T4=4a+c
....
Tn=an+c
here we found
a+c, 2a+c,3a+c,4a+c.......na+c
we also know according to Ap :-a sequence is called in Ap when common difference b/w two consecutive number is constant
here
d=a i.e constant
so this is Ap series and these common difference is a i.e. coefficient of n

Answer 2

Let there be a sequence whose nth term is a linear expression in n

an= An+B

an-1=A(n+1)+B

an+1-an=A

so,an+1-an is independent of n and is therefore a constant. So the sequence is an A.P. with common difference A