What is the sum of the first 8 terms of the geometric sequence 3, 12, 48?

The sum of a geometric sequence can be calculated using the formula:

S_n = a₁ × rac{(1 – r^n)}{1 – r}

Where:

• S_n is the sum of the first n terms.
• a₁ is the first term of the sequence.
• r is the common ratio between the terms.
• n is the number of terms.

In your case:

• The first term, a₁, is 3.
• The second term is 12, and the third term is 48.

To find the common ratio r, we divide the second term by the first term:

r = 12 / 3 = 4

We can confirm this by dividing the third term by the second term:

r = 48 / 12 = 4

So, r = 4, and the common ratio is consistent throughout the sequence.

Now, let’s calculate the sum of the first 8 terms (n = 8):

S₈ = 3 × rac{(1 – 4^8)}{1 – 4}

Calculating 4^8:
4⁸ = 65536

Now substituting this back into the sum formula:

S₈ = 3 × rac{(1 – 65536)}{-3}

Simplifying further:

S₈ = 3 × rac{-65535}{-3}

The -3 cancels out, leading to:

S₈ = 65535

Therefore, the sum of the first 8 terms of the geometric sequence is 65535.