Understanding Arithmetic Sequences
Arithmetic sequences are a special kind of math list where each number is added to or taken away from by the same amount. This amount is called the common difference, which we can write as (d).
Here’s how we think about the numbers in an arithmetic sequence:
Common Difference: The gap between each number in the list is always the same. We can find it like this:
(d = a_{n} - a_{n-1})
Explicit Formula: We can write any number in the sequence using this formula:
(a_n = a_1 + (n-1)d)
Sum of Terms: If you want to add up the first (n) numbers in the sequence, you can use this formula:
(S_n = \frac{n}{2}(a_1 + a_n))
Linear Growth: When you draw an arithmetic sequence on a graph, it makes a straight line. This shows that the numbers grow at a constant speed.
Real-Life Uses: These sequences can help in everyday situations, like figuring out how much money you save over time or how far you travel if you move at a steady pace.
Understanding Arithmetic Sequences
Arithmetic sequences are a special kind of math list where each number is added to or taken away from by the same amount. This amount is called the common difference, which we can write as (d).
Here’s how we think about the numbers in an arithmetic sequence:
Common Difference: The gap between each number in the list is always the same. We can find it like this:
(d = a_{n} - a_{n-1})
Explicit Formula: We can write any number in the sequence using this formula:
(a_n = a_1 + (n-1)d)
Sum of Terms: If you want to add up the first (n) numbers in the sequence, you can use this formula:
(S_n = \frac{n}{2}(a_1 + a_n))
Linear Growth: When you draw an arithmetic sequence on a graph, it makes a straight line. This shows that the numbers grow at a constant speed.
Real-Life Uses: These sequences can help in everyday situations, like figuring out how much money you save over time or how far you travel if you move at a steady pace.