Sasha Sets A Goal To Read 5 Minutes Longer Than Each Previous Day For 30 Days. On The First Day, Sasha Reads For 20 Minutes. The Expression $\sum_{n=1}^{30}[20+5(n-1)] Represents The Total Number Of Minutes Sasha Reads During The 30 Days. How

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Introduction

In this article, we will delve into the world of mathematics and explore a real-life scenario where an individual sets a goal to read for a certain number of minutes each day, with the goal of increasing the reading time by 5 minutes every day for 30 days. We will use the mathematical expression ∑n=130[20+5(n−1)]\sum_{n=1}^{30}[20+5(n-1)] to represent the total number of minutes Sasha reads during the 30 days.

The Problem

Sasha sets a goal to read 5 minutes longer than each previous day for 30 days. On the first day, Sasha reads for 20 minutes. The expression ∑n=130[20+5(n−1)]\sum_{n=1}^{30}[20+5(n-1)] represents the total number of minutes Sasha reads during the 30 days. We need to find the value of this expression.

Breaking Down the Expression

The given expression can be broken down into two parts: the constant term 20 and the variable term 5(n−1)5(n-1). The constant term 20 represents the initial reading time of 20 minutes on the first day. The variable term 5(n−1)5(n-1) represents the additional reading time of 5 minutes each day, starting from the second day.

Simplifying the Expression

To simplify the expression, we can start by evaluating the variable term 5(n−1)5(n-1). This term can be rewritten as 5n−55n - 5, which represents the additional reading time of 5 minutes each day.

Evaluating the Summation

The given expression can be rewritten as ∑n=130[20+5n−5]\sum_{n=1}^{30}[20+5n-5]. This expression represents the total number of minutes Sasha reads during the 30 days.

Using the Formula for the Sum of an Arithmetic Series

The expression ∑n=130[20+5n−5]\sum_{n=1}^{30}[20+5n-5] can be evaluated using the formula for the sum of an arithmetic series. The formula is given by:

∑n=1N(a+nd)=N2[2a+(N−1)d]\sum_{n=1}^{N} (a + nd) = \frac{N}{2} [2a + (N-1)d]

where aa is the first term, dd is the common difference, and NN is the number of terms.

Applying the Formula

In this case, the first term aa is 20, the common difference dd is 5, and the number of terms NN is 30. Plugging these values into the formula, we get:

∑n=130[20+5n−5]=302[2(20)+(30−1)5]\sum_{n=1}^{30} [20+5n-5] = \frac{30}{2} [2(20) + (30-1)5]

Simplifying the Expression

Simplifying the expression, we get:

∑n=130[20+5n−5]=15[40+145]\sum_{n=1}^{30} [20+5n-5] = 15 [40 + 145]

Evaluating the Expression

Evaluating the expression, we get:

∑n=130[20+5n−5]=15[185]\sum_{n=1}^{30} [20+5n-5] = 15 [185]

Final Answer

The final answer is:

∑n=130[20+5n−5]=2775\sum_{n=1}^{30} [20+5n-5] = 2775

Conclusion

In this article, we explored a real-life scenario where an individual sets a goal to read for a certain number of minutes each day, with the goal of increasing the reading time by 5 minutes every day for 30 days. We used the mathematical expression ∑n=130[20+5(n−1)]\sum_{n=1}^{30}[20+5(n-1)] to represent the total number of minutes Sasha reads during the 30 days. We broke down the expression, simplified it, and evaluated it using the formula for the sum of an arithmetic series. The final answer is 2775.

Additional Resources

For more information on the sum of an arithmetic series, please refer to the following resources:

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Introduction

In our previous article, we explored a real-life scenario where an individual sets a goal to read for a certain number of minutes each day, with the goal of increasing the reading time by 5 minutes every day for 30 days. We used the mathematical expression ∑n=130[20+5(n−1)]\sum_{n=1}^{30}[20+5(n-1)] to represent the total number of minutes Sasha reads during the 30 days. In this article, we will answer some frequently asked questions related to Sasha's reading challenge.

Q&A

Q: What is the initial reading time on the first day?

A: The initial reading time on the first day is 20 minutes.

Q: How much does Sasha increase her reading time each day?

A: Sasha increases her reading time by 5 minutes each day.

Q: What is the total number of minutes Sasha reads during the 30 days?

A: The total number of minutes Sasha reads during the 30 days is 2775.

Q: How did you evaluate the expression ∑n=130[20+5(n−1)]\sum_{n=1}^{30}[20+5(n-1)]?

A: We evaluated the expression using the formula for the sum of an arithmetic series.

Q: What is the formula for the sum of an arithmetic series?

A: The formula for the sum of an arithmetic series is given by:

∑n=1N(a+nd)=N2[2a+(N−1)d]\sum_{n=1}^{N} (a + nd) = \frac{N}{2} [2a + (N-1)d]

where aa is the first term, dd is the common difference, and NN is the number of terms.

Q: How did you simplify the expression ∑n=130[20+5n−5]\sum_{n=1}^{30}[20+5n-5]?

A: We simplified the expression by evaluating the variable term 5(n−1)5(n-1) and then applying the formula for the sum of an arithmetic series.

Q: What is the significance of the number 2775 in this context?

A: The number 2775 represents the total number of minutes Sasha reads during the 30 days.

Q: Can you provide more information on the sum of an arithmetic series?

A: Yes, please refer to the following resources for more information on the sum of an arithmetic series:

Conclusion

In this article, we answered some frequently asked questions related to Sasha's reading challenge. We provided information on the initial reading time, the increase in reading time each day, the total number of minutes read during the 30 days, and the evaluation of the expression ∑n=130[20+5(n−1)]\sum_{n=1}^{30}[20+5(n-1)]. We also provided information on the formula for the sum of an arithmetic series and the significance of the number 2775 in this context.

Additional Resources

For more information on the sum of an arithmetic series, please refer to the following resources:

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