7. Practice Problems

1. Find the 6th term of the sequence: 2, 6, 18, 54,…

To find the 6th term of a geometric sequence, we first need to identify the first term (𝑎) and the common ratio (𝑟). In this case, the first term is 2. We can find the common ratio by dividing the second term by the first term: 6 / 2 = 3. Thus, each term is obtained by multiplying the previous term by 3. We can find the 6th term using the formula: T = a * r^(n-1). Therefore, T = 2 * 3^(6-1) = 2 * 243 = 486.

Think of a plant that triples its height every week. If it starts at 2 cm in the first week, after 6 weeks, its height can be calculated using the same rules as the 6th term of the sequence, giving us a tangible connection to how growth works over time.

2. Determine the sum of the first 7 terms: 1, 1/2, 1/4,…

To determine the sum of the first 7 terms in a geometric sequence, we need to identify the first term (𝑎) and the common ratio (𝑟). Here, the first term is 1 and the common ratio is 1/2, since each term is half of the previous one. We then use the sum formula for finite geometric series: S = a * (1 - r^n) / (1 - r), where n is the number of terms. Thus, S = 1 * (1 - (1/2)^7) / (1 - 1/2). After calculation, we find the sum.

Imagine you are sharing a pizza with 7 friends, and with each sharing round, you take half of the remaining pizza each time. Calculating how much pizza you’ve eaten after 7 rounds can help students visualize the diminishing returns and sums of geometric sequences.

3. Is the sequence 2, 5, 10, 17 geometric? Justify.

To determine if a sequence is geometric, we need to check if the ratio of successive terms is constant. We calculate the ratios: 5/2, 10/5, and 17/10. If these ratios are the same, the sequence is geometric. In this case, the ratios are not equal (2.5, 2, 1.7), indicating that this is not a geometric sequence.

It's like checking if a set of stairs is uniform. If each step is the same height (constant ratio), then they are part of a uniform series; if not, they represent different heights—just like this sequence.

4. If 𝑎 = 4, 𝑟 = 1/4, find the sum of the infinite series.

To find the sum of an infinite geometric series, we need the first term (𝑎) and the common ratio (𝑟). In this case, 𝑎 is 4 and 𝑟 is 1/4. We use the formula for an infinite series: S = a / (1 - r), applicable only if |𝑟| < 1, which it is here. We calculate S = 4 / (1 - 1/4) = 4 / (3/4) = 4 * (4/3) = 16/3.

This can be likened to money being added to your savings account every month, where the first deposit is $4 and each month you add a quarter of the remaining previous balance. Even if the amounts reduce, they form a sum that can be calculated using our geometric series formula.

5. The population of bacteria triples every hour. Starting from 200, what is the population after 5 hours?

In this problem, we need to find the population after 5 hours, knowing that the bacteria triple every hour. The first term is 200, and the common ratio is 3. We can use the formula for the n-th term: T = a * r^(n-1). So for 5 hours, T = 200 * 3^(5) = 200 * 243 = 48600. Thus, after 5 hours, the bacteria population grows to 48,600.

Imagine a video game where every hour, your character's health triples. Starting from 200 health points, how powerful would your character be after 5 hours? This helps students visualize exponential growth as they relate to progress in gaming.

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