Solving the B Sequence: Tips and Tricks for Finding the Closed-Form Solution

[mknabster]

This sequence is stumping me, how would I go about solving this?

Write the sequence B = 1, -1/2, +1/4, -1/8, +1/16, ... in closed-form. I tried using (-1) has the top with different variable son the bottom, but nothing seems to add up. The 1 in the front is throwing me off. Any help would be appreciated. thank you

 

[mathman]

(-1/2)ⁿ, where n starts from 0.

 

[mknabster]

Oh so you do include 0 in the number line. HOw does this look: 1/(-2)ⁿ

 

[mathman]

Oh so you do include 0 in the number line. HOw does this look: 1/(-2)ⁿ

It looks the same. -1/2 = 1/(-2)

Also (a/b)ⁿ = aⁿ/bⁿ

 

[blue_raver22]

I suggest approaching this problem by first understanding the pattern of the sequence. Each term alternates between positive and negative, and the denominators are powers of 2. This suggests that the sequence may be related to the geometric series formula: a + ar + ar^2 + ar^3 + ... = a/(1-r).

In this case, the first term (a) is 1 and the common ratio (r) is -1/2. Plugging these values into the formula, we get B = 1/(1-(-1/2)) = 1/(3/2) = 2/3.

Therefore, the closed-form solution for this sequence is B = (2/3)^n, where n is the position of the term in the sequence (starting from n=0).

I recommend verifying this solution by plugging in different values for n and seeing if they match the given sequence. Additionally, you can use mathematical induction to prove that this closed-form solution holds for all values of n.

I hope this helps in solving the B sequence and understanding the importance of recognizing patterns in mathematical problems. Remember to always approach problems with a systematic and logical mindset. Good luck!