What is the sum of the geometric sequence 4, 24, and 144 if there are 7 terms?

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To find the sum of the first 7 terms of the geometric sequence that starts with 4, 24, and 144, we first need to determine the common ratio of the sequence.

The common ratio (r) can be found by dividing the second term (24) by the first term (4):

r = 24 / 4 = 6

Next, we can confirm the common ratio by dividing the third term (144) by the second term (24):

r = 144 / 24 = 6

With the first term (a = 4) and the common ratio (r = 6), we can calculate the sum of the first n terms of a geometric sequence using the formula:

S_n = a \frac{(1 - r^n)}{1 - r}

Here:

  • S_n = the sum of the first n terms
  • a = the first term (4)
  • r = the common ratio (6)
  • n = the number of terms (7)

Substituting the values into the formula:

S_7 = 4 \frac{(1 - 6^7)}{1 - 6}

First, calculate 6^7:

6^7 = 279936

Now plug this back into the formula:

S_7 = 4 \frac{(1 - 279936)}{-5}

This simplifies to:

S_7 = 4 \frac{-279935}{-5} = 4 \times 55987 = 223948

Thus, the sum of the first 7 terms of the geometric sequence is 223948.

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