The second term of a geometric sequence is 6 and the fourth term is 5
Find the sums of two possible series to 20 terms.

Question

The second term of a geometric sequence is 6 and the fourth term is 5 Find the sums of two possible series to 20 terms.

UpStudy AI · Accepted Answer

Let's analyze the given geometric sequence to find the sums of the two possible 20-term series.

### Given:
- **Second term**: $$ ar = 6 $$
- **Fourth term**: $$ ar^3 = 5 $$

### Step 1: Find the Common Ratio $$ r $$ and First Term $$ a $$

From the second and fourth terms:
$$
\frac{ar^3}{ar} = r^2 = \frac{5}{6} \implies r = \pm \sqrt{\frac{5}{6}} \approx \pm 0.91287
$$
 
Now, solve for $$ a $$:
$$
ar = 6 \implies a = \frac{6}{r} = \frac{6}{\pm \sqrt{\frac{5}{6}}} = \pm \frac{6\sqrt{6}}{\sqrt{5}} \approx \pm 6.570
$$

### Step 2: Calculate the Sum of 20 Terms for Each Possible Series

The sum of the first $$ n $$ terms of a geometric series is given by:
$$
S_n = a \frac{1 - r^n}{1 - r}
$$

#### **Case 1:** $$ r = \sqrt{\frac{5}{6}} \approx 0.91287 $$
$$
a \approx \frac{6}{0.91287} \approx 6.570
$$
$$
S_{20} = 6.570 \times \frac{1 - (0.91287)^{20}}{1 - 0.91287} \approx 63.4
$$

#### **Case 2:** $$ r = -\sqrt{\frac{5}{6}} \approx -0.91287 $$
$$
a \approx \frac{6}{-0.91287} \approx -6.570
$$
$$
S_{20} = -6.570 \times \frac{1 - (-0.91287)^{20}}{1 - (-0.91287)} \approx -2.88
$$

### Conclusion:
The sums of the two possible 20-term geometric series are:

- **Approximately 63.4** when the common ratio is positive.
- **Approximately -2.88** when the common ratio is negative.

### Rounded Answer:
After simplifying, the two possible 20-term sums are approximately 63.4 and –2.88.
The sums of the two possible 20-term geometric series are approximately 63.4 and –2.88.

The second term of a geometric sequence is 6 and the fourth term is 5
Find the sums of two possible series to 20 terms.

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