What is the sum of the geometric sequence 1, 6, 36 if there are 7 terms?

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A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. In this case, we can identify the terms of the given sequence: 1, 6, and 36.

First, let’s determine the common ratio. We can find it by dividing the second term by the first term:

r = 6 / 1 = 6

We can verify this ratio by dividing the third term by the second term:

r = 36 / 6 = 6

Both calculations confirm that the common ratio (r) is 6.

Next, we need to find the sum of the first 7 terms in this geometric sequence. The formula for the sum of the first n terms (Sn) of a geometric sequence is given by:

Sn = a * (1 - rn) / (1 - r)

Here,

  • a is the first term, which is 1.
  • r is the common ratio, which is 6.
  • n is the total number of terms, which is 7.

Now, substituting these values into the formula:

S7 = 1 * (1 - 67) / (1 - 6)

Calculating 67 first:

67 = 279936

Now substituting it back into our sum formula:

S7 = 1 * (1 - 279936) / (1 - 6)

This simplifies to:

S7 = (1 - 279936) / (-5)

Which further simplifies to:

S7 = -279935 / -5 = 55987

So, the sum of the first 7 terms of the geometric sequence is 55987.

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