Compare arithmetic and geometric sequences

• Distinguish a constant first difference from a constant nonzero ratio.
• Write explicit nth-term formulas with the correct n minus one exponent or multiplier.
• Check a formula against several listed terms before using it for prediction.

Build the idea

Worked example

1. Step 1

   The differences are not constant, but every term divided by its predecessor equals three. The sequence is geometric with first term five and ratio three.

   $$\frac{15}{5}=\frac{45}{15}=\frac{135}{45}=3$$

2. Step 2

   Insert the first term and ratio into the geometric template. The exponent n minus one makes the first exponent zero.

   $$g_n=5\cdot3^{n-1}$$

3. Step 3

   For the sixth term, the ratio has been applied five times. Three to the fifth is two hundred forty-three, and multiplying by five gives one thousand two hundred fifteen.

   $$g_6=5\cdot3^5=1215$$

• Using the formula with exponent n instead of n minus one.

  The first term must use zero repeated ratio steps. Substitute n equals one to catch an off-by-one exponent immediately.

• Classifying a sequence from only one pair of consecutive terms.

  Check the difference or ratio across every available adjacent pair because a partial pattern may not continue.

Classify twelve, eight, four, zero, write an explicit formula, and find the tenth term.

This reviewed reading order covers the lesson explanation, equations, example, and checks without claiming that a video exists.

1. Part 1

   Lesson overview

   Arithmetic sequences add a constant difference, while geometric sequences multiply by a constant ratio. Their explicit formulas encode linear and exponential growth. A sequence attaches an index to each term. Arithmetic change adds the same amount at every step, so the term graph lies on discrete points of a line. Distinguish a constant first difference from a constant nonzero ratio.

2. Part 2

   Visual model and equations

   Geometric change multiplies by the same ratio. Repeating that multiplication creates powers, so the explicit formula uses the ratio raised to the quantity n minus one. Show an arithmetic bar growing by three identical blocks each step beside a geometric bar doubling its entire length. One pattern adds a fixed piece; the other scales the accumulated whole. The nth arithmetic term equals the first term plus the quantity n minus one times the common difference. The nth geometric term equals the first term times the common ratio raised to the quantity n minus one.

   • The nth arithmetic term equals the first term plus the quantity n minus one times the common difference.
   • The nth geometric term equals the first term times the common ratio raised to the quantity n minus one.
3. Part 3

   Worked example

   Classify the sequence five, fifteen, forty-five, one hundred thirty-five, write its explicit formula, and find the sixth term. Compare consecutive differences and ratios, use the stable operation to choose a model, then evaluate that model at n equals six. The differences are not constant, but every term divided by its predecessor equals three. The sequence is geometric with first term five and ratio three. Insert the first term and ratio into the geometric template. The exponent n minus one makes the first exponent zero. For the sixth term, the ratio has been applied five times. Three to the fifth is two hundred forty-three, and multiplying by five gives one thousand two hundred fifteen. The sequence is geometric, its explicit formula is five times three raised to the quantity n minus one, and its sixth term is one thousand two hundred fifteen. The formula gives five at n equals one, fifteen at n equals two, and forty-five at n equals three, matching the listed terms before the prediction is used.

   • Fifteen over five, forty-five over fifteen, and one hundred thirty-five over forty-five all equal three.
   • The nth term equals five times three raised to the quantity n minus one.
   • The sixth term equals five times three to the fifth, which is one thousand two hundred fifteen.
4. Part 4

   Checks, practice, and scope

   The n minus one appears because no change has occurred at the first term. Testing n equals one should always reproduce the initial value before a formula is trusted. Using the formula with exponent n instead of n minus one. The first term must use zero repeated ratio steps. Substitute n equals one to catch an off-by-one exponent immediately. Classifying a sequence from only one pair of consecutive terms. Check the difference or ratio across every available adjacent pair because a partial pattern may not continue. Classify twelve, eight, four, zero, write an explicit formula, and find the tenth term. The consecutive differences are all negative four, so use the arithmetic formula with first term twelve. The sequence is arithmetic with a sub n equal to twelve minus four times the quantity n minus one, and the tenth term is negative twenty-four. Constant difference signals arithmetic change; constant ratio signals geometric change. The n minus one count records the number of changes after the first term. Checking early indices protects predictions from classification and indexing errors. Not every sequence is arithmetic or geometric, and a short finite list can fit multiple more complicated rules. A zero term can make consecutive ratios undefined, so geometric classification must be handled carefully.

Scope and limitations

• Not every sequence is arithmetic or geometric, and a short finite list can fit multiple more complicated rules.
• A zero term can make consecutive ratios undefined, so geometric classification must be handled carefully.