The producity of a worker at a scrapbook sticker company

Question

1. Find the next four terms of the recursively-defined sequence:

2. Let a₀, a₁, a₂,... be defined by the formula a_n = 5_n-1, for all integers n ≥ 0. Prove by induction that this sequence is a solution to the recurrence relation a_k = a_k-1 + 5, for all integers k ≥ 1 where a₀ = -1.

3. f(a) =  _k=1Σ⁴ ka^k. Express this polynomial in a recursively.

4. The producity of a worker at a scrapbook sticker company increased as an Arithmetic Sequence (Progression) over a period of 7 days. That is, on day n she produced a_n = a₁ + (n - 1)d stickers. If a₇ = 160 and d = 10 what is the total number of stickers she produced over the period of 7 days?

5. Tom decided to do an exercise involving the Fibonacci and Geometric sequences. He will be pouring liquid into a container. For the first 10 days he will pour in liters of liquid corresponding to the Fibonacci sequence, starting with 1 liter on the first day and 1 liter on the second day. He will use day 10 as the first day of his Geometric sequence and for the next 3 days he will increase the number of liters he pours in by 10% of the previous day. How many liters will he pour into the container on day 13?

6. Which of the following are second-order linear homogeneous recurrence relations with constant coefficients?  Place "yes" against all that are. If anyone is not give a reason.

a. a_k = ka_k-1 - a_k-2 

b. b_k = b_k-1 + 2b_k-2 

c. c_k = c_k-1 - c_k-2²

d. d_k = d_k-1 - πd_k-2 

e. r_k = r_k-1 - 2r_k-2 + 2  

f. s_k = 10s_k-2   

g. u_k = 2u_k-2  

6. Find a closed form expression that is a solution to the sequence defined by the following recurrence elation and initial conditions (use the method of characteristic equation):

7. Find a solution for the following recurrence relation and initial conditions (use the method of characteristic equation).

Answer 1

For all the questions illustrated below, find out the answers by methodically solving it and please show the computation part. Q1. The productivity of a worker at a scrapbook sticker company increased as the Arithmetic Sequence (Progression) over a time-period of 7 days. That is, on day n she generated an = a1 + (n - 1)d stickers. If a7 = 160 and d = 10 determine the total number of stickers she generated over the period of 7 days? Q2. Tom decided to do an exercise engaging the Fibonacci and Geometric sequences. He will be pouring liquid to a vessel. For the primary 10 days he will pour in liters of liquid respective to the Fibonacci sequence, beginning with 1 liter on the first day and 1 liter on the second day. He will employ day 10 as the first day of his Geometric sequence and for the subsequent 3 days he will raise the number of liters he pours in by 10% of the preceding day. Determine how many liters will he pour to the container on day 13?