What is the sum of the geometric sequence 3, 15, 75 if there are 7 terms?

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To find the sum of a geometric sequence, we first need to identify the first term and the common ratio. In the sequence you provided:

  • First term (a) = 3
  • Second term = 15
  • Third term = 75

Now, we can find the common ratio (r). The common ratio can be found by dividing any term by its preceding term:

  • r = 15 / 3 = 5
  • r = 75 / 15 = 5

So, we have the first term a = 3 and the common ratio r = 5.

The formula for the sum of the first n terms of a geometric series is:

Sn = a * (1 – rn) / (1 – r)

Here, we want to find the sum of the first 7 terms (:n = 7):

Plugging in the values:

S7 = 3 * (1 – 57) / (1 – 5)

Calculating the parts:

  • 57 = 78125
  • So, S7 = 3 * (1 – 78125) / (1 – 5)
  • S7 = 3 * (-78124) / (-4)
  • S7 = 3 * 19531 = 58593

Thus, the sum of the first 7 terms of the geometric sequence 3, 15, 75 is 58593.

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