use the forward substitution method to obtain the closed form formula for the recurrence relation

an = 2an-1 +1, and a0 = 0.

Step-by-step explanation:

We are given the following information:

We will now evaluate values for n =1, 2, 3, 4, 5 and so on.

By forward substitution method:

If we continue in this manner, we can see a general trend and we can say that

This is the solution for given recurrence relation.

hello from mrbilldoesmath!

answer:

(7 x + 6 y) (49 x^2 - 42 x y + 36 y^2)

discussion:

the trick is to recognize that 343 = 7^3 and 216 = 6^3 and, in general,

a^3 + b^3 =   (a + b) (a^2 - a b + b^2).

in our case

343x^3+216y^3   =   (7x)^3 + (6y)^3

so take a = 7x and b = 6y in the above equation to get the answer.

you,

mrb