Question

for each sequence, determine whether it appears to be arithmetic, ge 14, 12, 10, 8, ... ∘ arithmetic ∘ geometric ∘ neither 25, 36, 49, 64, ... ∘ arithmetic ∘ geometric ∘ neither 1600, 400, 100, 25, ... ∘ arithmetic ∘ geometric ∘ neither

Explanation:

Step1: Check first sequence (arithmetic)

Calculate consecutive differences:
$12-14=-2$, $10-12=-2$, $8-10=-2$
Common difference exists.

Step2: Check first sequence (geometric)

Calculate consecutive ratios:
$\frac{12}{14}=\frac{6}{7}$, $\frac{10}{12}=\frac{5}{6}$
Ratios are not equal.

Step3: Check second sequence (arithmetic)

Calculate consecutive differences:
$36-25=11$, $49-36=13$, $64-49=15$
Differences are not equal.

Step4: Check second sequence (geometric)

Calculate consecutive ratios:
$\frac{36}{25}=1.44$, $\frac{49}{36}\approx1.36$
Ratios are not equal.

Step5: Check third sequence (arithmetic)

Calculate consecutive differences:
$400-1600=-1200$, $100-400=-300$
Differences are not equal.

Step6: Check third sequence (geometric)

Calculate consecutive ratios:
$\frac{400}{1600}=\frac{1}{4}$, $\frac{100}{400}=\frac{1}{4}$, $\frac{25}{100}=\frac{1}{4}$
Common ratio exists.

Answer:

1. 14, 12, 10, 8, ...: Arithmetic
2. 25, 36, 49, 64, ...: Neither
3. 1600, 400, 100, 25, ...: Geometric