Obtain the generating function for the sequence 02, 12,22, 32,42…..

To obtain the generating function for the sequence 02, 12, 22, 32, 42..., we can follow these steps:

Step 1: Write down the sequence in a general form. The given sequence can be written as an = 10n + 2, where n is the position of the term in the sequence.

Step 2: Express the general form in terms of a variable. Let's use x as the variable to express the general form. So, an = 10n + 2 becomes an = 10x + 2.

Step 3: Write down the generating function. The generating function for the sequence can be written as G(x) = a0 + a1x + a2x^2 + a3x^3 + ..., where a0, a1, a2, a3, ... are the terms of the sequence.

Step 4: Substitute the general form into the generating function. Substituting the general form an = 10x + 2 into the generating function, we get: G(x) = (10(0) + 2) + (10(1) + 2)x + (10(2) + 2)x^2 + (10(3) + 2)x^3 + ...

Simplifying this expression, we have: G(x) = 2 + 12x + 22x^2 + 32x^3 + ...

Therefore, the generating function for the sequence 02, 12, 22, 32, 42... is G(x) = 2 + 12x + 22x^2 + 32x^3 + ...