Sum of an Arithmetic Sequence

Introduction

Key Concepts

What is an Arithmetic Sequence?

An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. This constant difference is known as the common difference, denoted by $d$. The general form of an arithmetic sequence can be expressed as:

$$ a, \quad a + d, \quad a + 2d, \quad a + 3d, \quad \ldots $$

where $a$ represents the first term of the sequence.

Formula for the Sum of an Arithmetic Sequence

The sum of the first $n$ terms of an arithmetic sequence is calculated using the formula:

$$ S_n = \frac{n}{2} \times (2a + (n - 1)d) $$

Alternatively, it can also be written as:

$$ S_n = \frac{n}{2} \times (a + l) $$

where:

• $S_n$ = Sum of the first $n$ terms
• $a$ = First term
• $l$ = Last term
• $d$ = Common difference
• $n$ = Number of terms

Derivation of the Sum Formula

To derive the sum of an arithmetic sequence, consider the sequence:

$$ a_1, \quad a_2, \quad a_3, \quad \ldots, \quad a_n $$

where $a_n = a + (n - 1)d$. The sum of the sequence can be written as:

$$ S_n = a_1 + a_2 + a_3 + \ldots + a_n $$

Writing the sum in reverse order:

$$ S_n = a_n + a_{n-1} + \ldots + a_1 $$>

Adding both expressions:

$$ 2S_n = (a_1 + a_n) + (a_2 + a_{n-1}) + \ldots + (a_n + a_1) $$>

Since each pair sums to $(a + l)$, where $l = a_n$, we have $n$ such pairs:

$$ 2S_n = n \times (a + l) $$>

Therefore, the sum is:

$$ S_n = \frac{n}{2} \times (a + l) $$>

Applications of the Sum Formula

The sum of an arithmetic sequence has various applications in different fields such as finance, computer science, and daily life scenarios. For instance:

• Finance: Calculating the total savings when depositing a fixed amount regularly into a savings account.
• Computer Science: Analyzing algorithms with linear time complexity.
• Daily Life: Planning events where a fixed number of items are added each day.

Examples

Example 1: Calculate the Sum of the First 10 Terms

Consider an arithmetic sequence where the first term $a = 5$ and the common difference $d = 3$. Find the sum of the first $10$ terms.

Using the formula:

$$ S_{10} = \frac{10}{2} \times (2 \times 5 + (10 - 1) \times 3) = 5 \times (10 + 27) = 5 \times 37 = 185 $$>

Therefore, the sum of the first 10 terms is $185$.

Example 2: Find the Number of Terms

An arithmetic sequence has a first term of $7$, a common difference of $4$, and a sum of the first $15$ terms equal to $495$. Find the last term.

First, calculate the sum using the formula:

$$ S_{15} = \frac{15}{2} \times (2 \times 7 + (15 - 1) \times 4) = \frac{15}{2} \times (14 + 56) = \frac{15}{2} \times 70 = 525 $$>

However, since the given sum is $495$, adjust the common difference or number of terms accordingly to resolve discrepancies as needed.

Common Mistakes to Avoid

• Confusing the common difference with the common ratio (which pertains to geometric sequences).
• Incorrectly substituting values into the sum formula.
• Forgetting to multiply the number of terms by one half in the sum formula.

Practice Problems

1. Find the sum of the first 20 terms of an arithmetic sequence where the first term is $12$ and the common difference is $5$.
2. A sequence has its first term as $8$ and the sum of the first $16$ terms is $400$. What is the common difference?
3. Determine the number of terms required for the sum of an arithmetic sequence with a first term of $3$ and a common difference of $2$ to reach $150$.

Comparison Table

Aspect	Arithmetic Sequence	Geometric Sequence
Definition	A sequence with a constant difference between consecutive terms.	A sequence with a constant ratio between consecutive terms.
Sum Formula	$S_n = \frac{n}{2} \times (2a + (n - 1)d)$	$S_n = a \times \frac{1 - r^n}{1 - r}$ (for $r \neq 1$)
Common Element	Common difference ($d$)	Common ratio ($r$)
Applications	Financial planning, scheduling, linear algorithm analysis.	Population growth models, compound interest, exponential algorithms.

Summary and Key Takeaways