Question

NO LINKS!!

Part 1

For this problem, refer to the sequences graphed below
a. Identify each sequence as arithmetic, geometric, or neither
b. If it is arithmetic or geometric, describe the sequence generator


n t(n)
0 16
1 13
2 10
3 7
4 4
5 1

Answer 1

a) Arithmetic sequence

b) t(n) = 16 - 3n

Explanation:

An Arithmetic Sequence has a constant difference between each consecutive term.

A Geometric Sequence has a constant ratio (multiplier) between each consecutive term.

Part (a)

From inspection of the given table, t(n) decreases by 3 each time n increases by 1. Therefore, the sequence is arithmetic with a common difference of -3.

Part (b)

`\boxed{\begin{minipage}{8 cm}\underline{General form of an arithmetic sequence}\\\\$a_n=a+(n-1)d$\\\\where:\\\phantom{ww}$\bullet$ $a_n$ is the nth term. \\ \phantom{ww}$\bullet$ $a$ is the first term.\\\phantom{ww}$\bullet$ $d$ is the common difference between terms.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}`

Given:

• a = 13
• d = -3
• aₙ = t(n)

Substitute the initial value (when n = 1) and the common difference into the formula to create an equation for the nth term:

`\implies t(n)=13+(n-1)(-3)`

`\implies t(n)=13-3n+3`

`\implies t(n)=16-3n`

Answer 2

• a) Arithmetic,
• b) t(n) = 16 - 3n.

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We observe that the sequence is decreasing with common difference of - 3.

• 16, 13, 10, 7, 4, 1

Therefore, this is an AP, with the first term of t(1) = 13 and common difference of d = -3.

Use the nth term formula for AP:

• t(n) = t(1) + (n - 1)d
• t(n) = 13 + (n - 1)(- 3) = 13 - 3n + 3 = 16 - 3n