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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Solution

Given sequence with n th term defined by t_n= a + nb

So t_n-1= a + (n-1)b

⇒ t_n- t_n-1 = a + nb - a - b(n-1)

⇒ t_n- t_n-1 = b

Therefore, the difference between the preceding term and next term of this sequence is independent of 'n '.

Hence this sequence is an AP with common difference b.

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Consider the arithmetic sequence with first term and common difference. Is 1 a term of this sequence? What about 2?

Write the algebraic form of this sequence. Prove the no natural number occurs is this sequence.

Consider an arithmetic sequence whose m^thterm is 'n' and n^thterm is 'm'

a) Find the common difference of the sequence

b) Prove that ( m+n+p) ^thterm of the sequence is '-p'

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