Sure, let's consider the series where we start from 2 and add 2 to each term. For instance, if we want to find the 5th term of this series and also the sum of the series up to the 5th term, we can use the formulas mentioned earlier.
First, the formula for the nth term of an arithmetic series:
\[ a_n = a_1 + (n - 1) \cdot d \]
Given:
- \( a_1 = 2 \) (the first term)
- \( d = 2 \) (the common difference)
- \( n = 5 \) (the term number we want to find)
Let's substitute these values into the formula to find the 5th term:
\[ a_5 = 2 + (5 - 1) \cdot 2 \]
\[ a_5 = 2 + 4 \cdot 2 \]
\[ a_5 = 2 + 8 \]
\[ a_5 = 10 \]
So, the 5th term of the series is 10.
Now, let's find the sum of the series up to the 5th term using the formula for the sum of the first \( n \) terms of an arithmetic series:
\[ S_n = \frac{n}{2} \cdot (a_1 + a_n) \]
Given:
- \( n = 5 \)
- \( a_1 = 2 \)
- \( a_5 = 10 \)
Substituting these values into the formula:
\[ S_5 = \frac{5}{2} \cdot (2 + 10) \]
\[ S_5 = \frac{5}{2} \cdot 12 \]
\[ S_5 = \frac{5 \cdot 12}{2} \]
\[ S_5 = \frac{60}{2} \]
\[ S_5 = 30 \]
Therefore, the sum of the series up to the 5th term is 30.
The formula used to find the nth term of an arithmetic series is simply called the "nth term formula" or "general term formula" for an arithmetic sequence.
\[ a_n = a_1 + (n - 1) \cdot d \]
Where:
- \( a_n \) is the nth term of the series.
- \( a_1 \) is the first term of the series.
- \( n \) is the position of the term you want to find.
- \( d \) is the common difference between consecutive terms.
Similarly, the formula used to find the sum of the first \( n \) terms of an arithmetic series is known as the "sum of an arithmetic series formula."
\[ S_n = \frac{n}{2} \cdot (a_1 + a_n) \]
Where:
- \( S_n \) is the sum of the first \( n \) terms of the series.
- \( n \) is the number of terms.
- \( a_1 \) is the first term of the series.
- \( a_n \) is the nth term of the series.
These formulas are specific to arithmetic sequences and series and are derived from the properties of arithmetic progressions.
