Let us understand what is arithmetic sequence from an example. Suppose a woman purchased a new truck for $25,000. After 5 years, she estimates that she can sell the truck for $8,000. The total loss in value of the truck in 5 years will therefore be $17,000. This means that the truck will lose its value by $3,400 per year, for 5 years:
Truck is worth $25,000 at the time of purchase
Truck is worth $21,600 at the end of 1st year
Truck is worth $18,200 at the end of 2nd year
Truck is worth $14,800 at the end of 3rd year
Truck is worth $11,400 at the end of 4th year
Truck is worth $8,000 at the end of the 5th year
The values of the truck in the example are said to form an "arithmetic sequence", because they change by a constant amount of 1$3,400 each year, and the constant amount of change is called "common difference". In above example, the arithmetic sequence is: 25000, 21600, 18200, 14800, 11400, 8000, and the "common difference" is -3400.
There are 2 commonly used formulas for arithmetic sequence.
The nth term of an arithmetic sequence is
Where a1 is the 1st term, d is common difference, and an is the nth term.
The sum of the first n terms of the arithmetic sequence is
Where a1 is the 1st term, an is the nth term, and S is the sum
Geometric Sequence
A geometric sequence is a sequence of numbers that follows a pattern: the next term is found by multiplying the current term by a constant called the common ratio, r:
For example:
a1 = 2
a2 = 2 x 3 = 6
a3 = 6 x 3 = 18
a4 = 18 x 3 = 54
a5 = 54 x 3 = 162
To find the sum of the 1st n terms of the geometric sequence, you can use the following formula:
