OneNote Final Exam Review Monday November 28 2016 121 PM all

OneNote Final Exam Review Monday, November 28, 2016 1:21 PM all integers 6\' Convert to shapes HK Exit Full Screen Drawing 5

Solution

SOLUTION

Given h₀ = 2 and h_k = 4h_{k – 1} + 5,

h₁ = 4h₀ + 5

h₂ = 4h₁ + 5 = 4(4h₀ + 5) + 5 = 4²h₀ + 4x5 + 5

h₃ = 4h₂ + 5 = 4(4²h₀ + 4x5 + 5) + 5 = 4³h₀ + 4²x5 + 4x5 + 5

Thus, h₄ should be 4⁴h₀ + 4³x5 + 4²x5 + 4x5 + 5 or

In general, h_k = 4^kh₀ + 4^{k - 1}x5 + 4^{k - 2} x5 + …… + 4x5 + 5 ………… (1)    

The above analysis shows how to build up the recurrence relation.

In most such problems, the terms involving numerals can be further simplified to arrive at a concise formula. Again, in majority of cases, as in this case also, the terms form a GP.

4^{k - 1}x5 + 4^{k - 2} x5 + …… + 4x5 + 5 = 5(4^{k - 1} + 4^{k - 2} + …… + 4 + 1) = 5 x (a GP of k terms with a = 4^{k – 1} and r = 1/4) = 5[4^{k – 1}{1 – (1/4)^k}/{1 – (1/4)} = {(5x4)/3}[4^{k – 1}(4^k - 1)/4^k]

= (5/3) (4^k - 1) ……. (2)

(2) in (1): h_k = 4^kh₀ + (5/3) (4^k - 1) = 2x4^k [because h₀ = 2] + (5/3) 4^k – (5/3). So,

h_k = (11/3)4^k – (5/3) ANSWER or still better, 3h_k = 11.4^k - 5

[Some useful tips:

1. (4^{k - 1} + 4^{k - 2}  + …… + 4 + 1) has k terms and not (k - 1). This is a very popularly committed mistake.

2. Instead of summing as done above, reverse the series and then it becomes a GP of k terms with a = 1 and r =4, which is much easier and simpller to simplify]