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If nth term of a sequence is a linear expression in n then, prove that, the sequence, so formed, is an arithmetic sequence and the common difference if this arithmetic sequence is equal to the coefficient of n.
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This proves that the sequence is AP.
Recall that, in an AP,
T
which is the co-eff. of
n
in
T
n
.
Answered by
5
Suppose that,
Tn denotes the nth term of the sequence.
The general linear expression in n is an+b, where, a≠0.
We are given that, Tn=an+b;
n∈N, a,b∈R, a≠0.
Observe that,Tn+1−Tn={a(n+1)+b}−(an+b)=a=constant.
This proves that the sequence is AP.
Recall that, in an AP,
Tn+1−Tn is called the common difference (cd) of the AP.
Hence, the cd is a which is the co-eff. of n
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