How can you graph the six terms of a finite sequence where the first term is 5 and the common ratio is 125?

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To graph the six terms of a finite sequence, we first need to understand how the terms are generated based on the given information. In this case, the sequence starts with the first term, denoted as a1, which is equal to 5, and has a common ratio r equal to 125.

The formula for the nth term of a geometric sequence is:

an = a1 * r(n-1)

Using this formula, we can calculate the first six terms:

  1. Term 1: a1 = 5
  2. Term 2: a2 = 5 * 125(2-1) = 5 * 125 = 625
  3. Term 3: a3 = 5 * 125(3-1) = 5 * 1252 = 5 * 15625 = 78125
  4. Term 4: a4 = 5 * 125(4-1) = 5 * 1253 = 5 * 1953125 = 9765625
  5. Term 5: a5 = 5 * 125(5-1) = 5 * 1254 = 5 * 156250000 = 781250000
  6. Term 6: a6 = 5 * 125(6-1) = 5 * 1255 = 5 * 195312500000 = 976562500000

Now that we have our six terms: 5, 625, 78125, 9765625, 781250000, and 976562500000.

To graph these values, follow these steps:

  1. Set up a coordinate system where the x-axis represents the term number (from 1 to 6) and the y-axis represents the value of each term.
  2. Plot each of the points corresponding to the calculated values:
    • Point (1, 5)
    • Point (2, 625)
    • Point (3, 78125)
    • Point (4, 9765625)
    • Point (5, 781250000)
    • Point (6, 976562500000)

Due to the exponential growth of the sequence values, the graph will demonstrate a steep ascent, illustrating how quickly the terms increase as n increases. This characteristic of geometric sequences (especially with a high common ratio) results in a powerful visual representation of growth that is both informative and striking.

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