Prove that no matter what the real numbers a and b are,the sequence with nth term a+nb is always an A.P. What is the common difference?
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Step-by-step explanation:
Given the sequence which is defined by.
a
n
=a+nb.....eq(1)
and we know that nth term of an A.p is given by
a
n
=a
1
+(n−1)d
which can also be written as
a
n
=a
1
−d+nd.....eq(2)
comparing eq(1) and eq(2) we get
common difference is
d=b (by comparing coefficients of n)
and
a
1
−d=a
by putting value of d=b in above equation we get
a
1
−b=a
⟹a
1
=a+b (first term of A.P)
hence if a and b are real numbers this sequence form an A.P with first term a
1
=a+b
and have a common difference d=b
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