Question

What is the sum of the geometric sequence 1, 3, 9, ... if there are 10 terms? (5 points)

Answer 1

Answer:

$S_n = \frac{1 (1 - 3^{10})}{1 - 3} = 29524$

Step-by-step explanation:

There's a handy formula we can use to find the sum of a geometric sequence, and here it is

$S_n = \frac{a_1 (1 - r^n)}{1 - r}$

The value n represents the amount of terms you want to sum in the sequence. The variable r is known as the common ratio, and a is just some constant. Let's find those values.

First lets visualize this sequence

$n_1 = 1\\n_2 = 1 + 3\\n_3 = 1 + 3 + 3^2\\n_4=1+3+3^2+3^3\\...$

Okay so there's clearly a pattern here, let's write it a bit more concisely. For each n, starting at 1, we raise 3 to the (n-1) power, add it to what we had for the previous term.

$S_n = \sum{3^{n-1}} = 3^{1 - 1} + 3^{2 - 1} + 3^{3-1} ...$

Our coefficients of r, and a, are already here! As you can see below, r is just 3, and a is just 1.

$S_n = \sum{a*r^{n-1}}$

To finish up lets plug these coefficients in and get our sum after 10 terms.

$S_n = \frac{1 (1 - 3^{10})}{1 - 3} = 29524$