Question

b. is 119 a term in this sequence? explain how. 4. label represented sequence as arithmetic, geometric or neither. state next three terms a. 11, 7, 5, 2 c. 8, 4, 2, 1 b. ( t(n) = -2n + 7 ) d. 5, 5, 5, 5,

Explanation:

Part a: 11, 5, 1, 2

Step1: Check arithmetic pattern

Calculate differences: $5-11=-6$, $1-5=-4$, $2-1=1$. Differences are not constant.

Step2: Check geometric pattern

Calculate ratios: $\frac{5}{11}$, $\frac{1}{5}$, $\frac{2}{1}=2$. Ratios are not constant.

Step3: Classify and find next terms

Sequence is neither. No consistent pattern to define next terms.

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Part b: $f(n)=-2n+7$

Step1: Identify sequence type

This is a linear function, so it is arithmetic. Common difference $d=-2$.

Step2: Find first 4 terms

$n=1$: $f(1)=-2(1)+7=5$; $n=2$: $f(2)=-2(2)+7=3$; $n=3$: $f(3)=-2(3)+7=1$; $n=4$: $f(4)=-2(4)+7=-1$

Step3: Calculate next three terms

$n=5$: $f(5)=-2(5)+7=-3$; $n=6$: $f(6)=-2(6)+7=-5$; $n=7$: $f(7)=-2(7)+7=-7$

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Part c: 8, 4, 2, 1

Step1: Check geometric pattern

Calculate ratios: $\frac{4}{8}=\frac{1}{2}$, $\frac{2}{4}=\frac{1}{2}$, $\frac{1}{2}=\frac{1}{2}$. Common ratio $r=\frac{1}{2}$.

Step2: Classify sequence

Sequence is geometric.

Step3: Find next three terms

$1\times\frac{1}{2}=\frac{1}{2}$; $\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$; $\frac{1}{4}\times\frac{1}{2}=\frac{1}{8}$

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Part d: 5, 5, 5, 5

Step1: Check arithmetic pattern

Calculate differences: $5-5=0$ (constant). Common difference $d=0$.

Step2: Check geometric pattern

Calculate ratios: $\frac{5}{5}=1$ (constant). Common ratio $r=1$.

Step3: Classify and find next terms

Sequence is both arithmetic and geometric. Next terms are all 5.

Answer:

a. Classification: Neither; No definable next three terms due to inconsistent pattern
b. Classification: Arithmetic; Next three terms: $-3$, $-5$, $-7$
c. Classification: Geometric; Next three terms: $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$
d. Classification: Both arithmetic and geometric; Next three terms: $5$, $5$, $5$