Are the following sequences arithmetic or geometric? a) 4, 8, 16, 32, ... b) 35, 27, 19, 11, ... c) 5, 8, 11, 14, ...

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Understand the Problem

The question is asking to determine whether each of the provided sequences is arithmetic or geometric based on the definitions provided, focusing on the operations of addition/subtraction and multiplication/division.

Answer

a) Geometric b) Arithmetic c) Arithmetic
Answer for screen readers

a) Geometric sequence
b) Arithmetic sequence
c) Arithmetic sequence

Steps to Solve

  1. Identify the first sequence: 4, 8, 16, 32, ...

    Examine the differences and ratios between successive terms.

    • Difference between terms:
      • $8 - 4 = 4$
      • $16 - 8 = 8$
      • $32 - 16 = 16$

    The differences are not constant, so it's not arithmetic.

    • Ratios of terms:
      • $\frac{8}{4} = 2$
      • $\frac{16}{8} = 2$
      • $\frac{32}{16} = 2$

    The ratios are constant, indicating it's a geometric sequence with a common ratio of 2.

  2. Identify the second sequence: 35, 27, 19, 11, ...

    Again, analyze the differences:

    • Difference between terms:
      • $27 - 35 = -8$
      • $19 - 27 = -8$
      • $11 - 19 = -8$

    The differences are constant (-8), confirming it is an arithmetic sequence.

  3. Identify the third sequence: 5, 8, 11, 14, ...

    Check the differences to determine its type:

    • Difference between terms:
      • $8 - 5 = 3$
      • $11 - 8 = 3$
      • $14 - 11 = 3$

    The differences are constant (3), indicating it is also an arithmetic sequence.

a) Geometric sequence
b) Arithmetic sequence
c) Arithmetic sequence

More Information

  • The first sequence doubles each term, fitting the definition of a geometric sequence.
  • The second and third sequences subtract a constant value from each term, fitting the definition of an arithmetic sequence.

Tips

  • A common mistake is to confuse the difference between terms as being constant in a geometric sequence; they should be ratios instead.
  • The opposite can occur as well; one might look for constant ratios in an arithmetic sequence rather than checking the differences.

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