5. Practice Problems

1. Find the sum of the first 15 terms of the series: 5+10+15+…

This problem requires you to find the sum of the first 15 terms of an arithmetic series where the first term is 5 and the common difference is 5 (since each term increases by 5). The formula to find the sum of the first n terms (S_n) of an arithmetic series is given by:

S_n = (n/2) * (2a + (n-1)d)

Where:

• S_n is the sum of the first n terms.
• a is the first term.
• d is the common difference.
• n is the number of terms.

Plugging in our values: a = 5, d = 5, n = 15. So, using the formula:

S_15 = (15/2) * (2*5 + (15-1)*5) = (15/2) * (10 + 70) = (15/2) * 80 = 15 * 40 = 600.

Imagine you are saving money in a part-time job where each week you earn $5 more than the previous week. Week 1 you earn$5, week 2 you earn $10, and so on. If you continue this way for 15 weeks, you can calculate how much money you'll have saved in total using the arithmetic series formula.

2. A geometric series has first term 𝑎 = 4 and common ratio 𝑟 = . Find the sum of the first 6 terms.

In this problem, you need to calculate the sum of the first 6 terms of a geometric series. The formula for the sum of the first n terms of a geometric series when the common ratio r is not equal to 1 is:

S_n = a * (1 - r^n) / (1 - r)

Here, a is the first term (4), r is the common ratio, and n is 6. Since the common ratio is not specified in a numerical form, I'm assuming it reflects a mistake in the input. Let's say if it were 0.5, then:

S_6 = 4 * (1 - 0.5^6) / (1 - 0.5). The calculations would go as follows: = 4 * (1 - 1/64) / (1/2) = 4 * (63/64) / (1/2) = 8 * (63/64) = 63/8 = 7.875.

Consider a situation where you invest $4 in a business and every month, it doubles. In a month, you would have$4, the next month $8, and so on. By the end of 6 months, you can calculate your total investment return using the geometric series formula.

3. The 5th term of an arithmetic sequence is 20 and the 10th term is 35. Find the sum of the first 15 terms.

In this problem, we know two specific terms of the arithmetic sequence: the 5th term (20) and the 10th term (35). We can find the common difference (d) first by considering the relationship between these terms:

The nth term of an arithmetic sequence is defined as: t_n = a + (n-1)d.

Thus, For t_5: a + 4d = 20 For t_10: a + 9d = 35. Subtracting these equations gives: 5d = 15, leading to d = 3. Now that we have d, we can find a by plugging d back into either equation, for instance: 20 = a + 4*3 => a = 20 - 12 = 8. Now use a and d to find the sum of the first 15 terms: S_15 = (15/2) * (2a + (n-1)d) => S_15 = (15/2) * (16 + 42) = 15 * 29 = 435.

Think of this scenario where you receive an increase in your salary every year. In the 5th year, you earn $20,000, and in the 10th year, you earn$35,000. Understanding the arithmetic progression allows you to predict how much money you would have earned over the years and how the amount grows as you receive your increase.

4. Find 𝑛 if the sum of the first 𝑛 terms of the arithmetic series is 120, and the first term is 6 with a common difference of 2.

To solve for n, we will apply the arithmetic series sum formula: S_n = (n/2) * (2a + (n-1)d) We know S_n = 120, a = 6, and d = 2. Substituting the values:

120 = (n/2) * (2*6 + (n-1) * 2) This simplifies to: 120 = (n/2) * (12 + 2n - 2) 120 = (n/2) * (10 + 2n) Multiply both sides by 2 to eliminate the fraction: 240 = n * (10 + 2n), thus: 240 = 10n + 2n^2. Rearranging takes the form of quadratic: 2n^2 + 10n - 240 = 0. Factoring gives: n^2 + 5n - 120 = 0. Use the quadratic formula to solve for n: n = [-5 ± √(5^2 - 4 * 1 * (-120))] / (2 * 1).

Solving yields two potential values for n: 10 and -12. But since n must be positive, we find n = 10.

Imagine you are saving $6 every month and every month, you increase that by$2. You want to know how many months it will take to save $120. Using the arithmetic sequence principles, you can calculate and find out logically without enumerating every month.

5. Determine whether the following is an arithmetic or geometric sequence: 3,6,12,24,48

To classify the series, observe the pattern in the terms. For an arithmetic sequence, the difference between consecutive terms should remain constant. Let's calculate the differences: 6 - 3 = 3, 12 - 6 = 6, 24 - 12 = 12, 48 - 24 = 24. As you can see, the difference is not constant. Therefore, it is not an arithmetic sequence. Next, let's check if it's a geometric sequence by checking if the ratio between the terms is constant: 6/3 = 2, 12/6 = 2, 24/12 = 2, 48/24 = 2. Since each term doubles the previous term consistently, this series is a geometric sequence with a common ratio of 2.

Think of a scenario where you are doubling your savings every month. If this month you have $3, next month it’s$6, the month after that $12, then$24, and so forth. This reflects how you recognize patterns not just in math but in real life too.

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