Question

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

Write the algebraic form of this sequence. Prove that all natural numbers occur in this sequence.

Answer 1

First term of the given sequence = a + b =

Common difference = a =

Let 1 be the n^th term of the given sequence.

Thus, 1 is the 3^rd term of this sequence.

Now, let 2 be the m^th term of the given sequence.

Thus, 2 is the 7^th term of this sequence.

Therefore, 1 and 2 are the 3^rd and the 7^th terms of this sequence.

The algebraic form of the sequence is

Now, let 3 be the p^th term of the given sequence.

Thus, 3 is the 11^th term of this sequence.

∴ 3^rd term of the given sequence is 1, 7^th term is 2, 11^th term is 3 and so on.

Thus, the new sequence obtained is 3, 7, 11, …

Here, first term = a + b = 3

Common difference = a = 7 − 3 = 4

⇒ 4 + b = 3

⇒ b = 3 − 4 = −1

The algebraic form of the sequence is, where q is a natural number.

x_q will surely satisfy as x_q is the general form of the n^th term of this sequence which gives a natural number as the result.

As q is a natural number, so all the natural numbers occur in the sequence