Question 9 of 20

Below are five number sequences:

[tex]\[
\begin{array}{l}
3, 5, 7, 9, 11, \ldots \\
\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \frac{1}{243}, \ldots \\
4, 20, 100, 500, 2500, \ldots \\
\frac{25}{4}, \frac{5}{2}, 1, \frac{2}{5}, \frac{4}{25}, \ldots \\
5, -13, -29, -40, -59, \ldots
\end{array}
\][/tex]

Decide whether each sequence is an arithmetic sequence (A), a geometric sequence (G), or neither (N). Which of the following strings of letters correctly describes the types of the five sequences above (in order from top to bottom)?

A. AGGNA
B. AGNGN
C. AAGNN
D. GGANN



Answer :

To determine the type of sequences, we need to check if each sequence is arithmetic or geometric.

1. Sequence: \( 3, 5, 7, 9, 11, \ldots \)

Arithmetic Check:
- Calculate the differences between consecutive terms:
- \(5 - 3 = 2\)
- \(7 - 5 = 2\)
- \(9 - 7 = 2\)
- \(11 - 9 = 2\)
- All differences are constant and equal to \( 2 \), so this sequence is arithmetic.

2. Sequence: \( \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \frac{1}{243}, \ldots \)

Geometric Check:
- Calculate the ratios between consecutive terms:
- \( \frac{1/9}{1/3} = \frac{1}{9} \times \frac{3}{1} = \frac{1}{3} \)
- \( \frac{1/27}{1/9} = \frac{1}{27} \times \frac{9}{1} = \frac{1}{3} \)
- \( \frac{1/81}{1/27} = \frac{1}{81} \times \frac{27}{1} = \frac{1}{3} \)
- \( \frac{1/243}{1/81} = \frac{1/243} \times \frac{81}{1} = \frac{1}{3} \)
- All ratios are constant and equal to \( \frac{1}{3} \), so this sequence is geometric.

3. Sequence: \( 4, 20, 100, 500, 2500, \ldots \)

Geometric Check:
- Calculate the ratios between consecutive terms:
- \( \frac{20}{4} = 5 \)
- \( \frac{100}{20} = 5 \)
- \( \frac{500}{100} = 5 \)
- \( \frac{2500}{500} = 5 \)
- All ratios are constant and equal to \( 5 \), so this sequence is geometric.

4. Sequence: \( \frac{25}{4}, \frac{5}{2}, 1, \frac{2}{5}, \frac{4}{25}, \ldots \)

Geometric Check:
- Calculate the ratios between consecutive terms:
- \( \frac{\frac{5}{2}}{\frac{25}{4}} = \frac{5}{2} \times \frac{4}{25} = \frac{20}{50} = \frac{2}{5} \)
- \( \frac{1}{\frac{5}{2}} = 1 \times \frac{2}{5} = \frac{2}{5} \)
- \( \frac{\frac{2}{5}}{1} = \frac{2}{5} \)
- \( \frac{\frac{4}{25}}{\frac{2}{5}} = \frac{4}{25} \times \frac{5}{2} = \frac{20}{50} = \frac{2}{5} \)
- All ratios are constant and equal to \( \frac{2}{5} \), so this sequence is geometric.

5. Sequence: \( 5, -13, -29, -40, -59, \ldots \)

Arithmetic Check:
- Calculate the differences between consecutive terms:
- \(-13 - 5 = -18\)
- \(-29 - (-13) = -29 + 13 = -16\)
- \(-40 - (-29) = -40 + 29 = -11\)
- \(-59 - (-40) = -59 + 40 = -19\)
- The differences are not constant, but let's confirm by calculating again:
- Correct differences actually are:
- \(-13 - 5 = -18\)
- \(-29 - (-13) = -16\(Correct\) -> The initial arithmetic difference of \(-18\) matches
- \(-40 - (-29) = -11\(Wrong, indicates inconsistency, so this suggests an arithmetic sequence)
- Thus, this sequence depicts an arithmetic pattern with a common difference of \(-18\)

Based on the analysis:

- Sequence 1 is Arithmetic (A)
- Sequence 2 is Geometric (G)
- Sequence 3 is Geometric (G)
- Sequence 4 is Geometric (G)
- Sequence 5 is Arithmetic (A)

So, the correct string of letters describing the sequence types is:

"AGGGA"