Arithmetic Series - Sum of the Terms of Arithmetic Sequence

Summary

In this video, Teacher Turgon explains the concept of arithmetic series, focusing on two key formulas for calculating the sum of terms in an arithmetic sequence. The first formula is used when both the first and last terms are known, while the second formula is applied when the first term and common difference are given. He solves two example problems: one calculating the sum of the first 20 terms and another for the first 40 terms of different arithmetic sequences. The video offers a clear, step-by-step explanation to help viewers understand and apply these formulas.

Takeaways

• 📚 The video discusses arithmetic series and how to find the partial sum of a given number of terms in an arithmetic sequence.
• 📐 Two formulas for calculating the sum of an arithmetic series are introduced: one that uses the first and last term, and another that uses the first term and common difference.
• 🧮 Formula 1: Sₙ = n * (a₁ + aₙ) / 2, where n is the number of terms, a₁ is the first term, and aₙ is the last term.
• 📏 Formula 2: Sₙ = n/2 * (2a₁ + (n-1) * d), where d is the common difference, n is the number of terms, and a₁ is the first term.
• ✍️ Example 1: The sum of the first 20 terms in a sequence with a₁ = 5 and aₙ = 62 is calculated using Formula 1, resulting in Sₙ = 670.
• 📝 Simplification Tip: Instead of directly multiplying large numbers, simplify fractions to make calculations easier.
• 🔢 Example 2: The sum of the first 40 terms in an arithmetic series with a₁ = 2 and a common difference of 3 is calculated using Formula 2, resulting in Sₙ = 2420.
• 🧑‍🏫 A common difference is found by subtracting consecutive terms in the sequence. In Example 2, the difference is 3 (e.g., 5 - 2 = 3).
• 📊 Formula selection depends on whether the last term of the series is provided or if only the common difference and first term are given.
• 🎥 The video encourages viewers to like, subscribe, and hit the notification bell for updates.

Q & A

What is an arithmetic series?

An arithmetic series is the sum of a specific number of terms in an arithmetic sequence, where each term after the first is obtained by adding a constant value, called the common difference.

What does S sub n represent in the formulas?

S sub n represents the sum of the first 'n' terms in an arithmetic series.

What is the formula for the sum of an arithmetic series when the first and last terms are known?

The formula is S sub n = n * (a sub 1 + a sub n) / 2, where n is the number of terms, a sub 1 is the first term, and a sub n is the last term.

What is the alternative formula for the sum of an arithmetic series when the common difference is known?

The alternative formula is S sub n = n / 2 * (2 * a sub 1 + (n - 1) * d), where n is the number of terms, a sub 1 is the first term, and d is the common difference.

How do you choose between the two arithmetic series formulas?

You use the first formula when the first and last terms of the series are given. You use the second formula when you know the first term and the common difference, but not the last term.

In the first problem, what is the value of the sum of the first 20 terms when a sub 1 is 5 and a sub 20 is 62?

The sum of the first 20 terms is 670.

How was the sum of the first 20 terms in the first problem calculated?

The sum was calculated using the formula S sub n = n * (a sub 1 + a sub n) / 2. Substituting the values, it became S sub 20 = 20 * (5 + 62) / 2, which equals 670.

In the second problem, how was the common difference calculated?

The common difference was calculated by subtracting consecutive terms in the sequence. For example, 5 - 2 = 3, 8 - 5 = 3, and so on, giving a common difference of 3.

What is the sum of the first 40 terms in the sequence 2, 5, 8, 11...?

The sum of the first 40 terms is 2,420.

How was the sum of the first 40 terms in the second problem calculated?

The sum was calculated using the formula S sub n = n / 2 * (2 * a sub 1 + (n - 1) * d). Substituting the values, S sub 40 = 40 / 2 * (2 * 2 + 39 * 3), which results in 2,420.