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Arithmetic sequence geometric sequence recurrence sequence

1) Consider the sequence given by: U1 = 3.9, un+1 = un - 1.7 (n = 1,2,3,...).

i) What type of sequence is this?
ii) Please write the first four terms of the sequence.
iii) Find a closed form for the sequence.
iv) Use the closed form from 1. iii to find the value of n when un = -76

2) Consider the following geometric sequence.

400, 320, 256, 204.8, ...

i) Write down a recurrence system that describes this sequence.
(Denote the sequence by Xn, and its first term by x1.)

ii) Find a closed form for this sequence.
iii) Use the closed form part 2ii to find the tenth term of the sequence to four decimal places.

3) Consider the linear recurrence sequence

X1 = 23, xn+1 = 1.3xn - 12 (n = 1,2,3,...).

i) Find a closed form for the sequence.
ii) Use the closed form to find the eighth term of the sequence, correct to four significant figures.
iii) Describe the long-term behaviour of the sequence.

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Answers:

(1)

i. Arithmetic sequence

ii. u1 = 3.9, u2 = 2.2, u3 = 0.5, u4 = -1.2

iii.

u2 = u1 - 1.7

u3 = u2 - 1.7 = u1 - 1.7- 1.7 = u1 - 2(1.7)

This pattern continues, and we can write the nth term as,

un = 3.9 - (n-1) *1.7

iv.

un = -76 = u1 - (n-1) * 1.7

-76 = 3.9 - (n-1) * 1.7

Solve for n:

n = ...

Solution Summary

Three problems with several sub categories on arithmetic sequence, geometric sequence and recurrence sequence have been answered. I have provided very clear, step by step solution so that the student will understand the technique to solve similar problems in the future. Please download them as these are wonderful sample questions and answers on sequences and series.

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