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Here are a few recommended readings before getting started with this lesson.
Diego's father has just taken a loan from the National Bank to start his own company. His first monthly payment will be $2400, after which the monthly payments will decrease by $100 each month — in other words, he will pay $2300 the second month, $2200 the third, and so on, until he pays a total of $24500, including interest.
An arithmetic series is the sum of the terms of an arithmetic sequence. If the sequence is finite and short enough, calculating the sum of its terms is quite straightforward.
Example Arithmetic Sequence | 1,3,5,7,9 |
---|---|
Related Arithmetic Series | 1+3+5+7+9=25 |
However, if the sequence is still finite but longer, it can be tedious to add the terms by hand. In that case, the formula for an arithmetic sum can be used. If the arithmetic series is infinite, then the sum is said to diverge.
The applet shows the first five terms of a sum. Identify whether the given sum is an arithmetic series or not.
For a finite arithmetic sequence with n terms and general formula an=a1+(n−1)d, where a1 is the first term and d the common difference, the sum of all terms Sn can be calculated using the following formula.
Sn=2n(a1+an)
Write as a sum
Associative Property of Addition
Add terms
a1=6, d=6
Distribute 6
Subtract terms
Use the formula for the sum of an arithmetic series.
n=1
Identity Property of Multiplication
Add terms
Calculate the sum of all the terms of the given finite arithmetic series written in summation notation. Use the formula for the sum of an arithmetic series.
A vicious gang of thieves, famous for the masks they wear, is concocting a scheme to steal the National Bank's gold reserve.
Before they make their move, they need to know how many bars of gold are kept in the reserve. According to the gang's mastermind, the bars of gold are stacked in the form of a triangle where each row has one more bar than the previous one.
a1=1, d=1
Identity Property of Multiplication
Remove parentheses
Subtract term
a1=20, d=1
Identity Property of Multiplication
Remove parentheses
Subtract term
a1=50, d=4
Distribute 4
Subtract term
a1=50, a15=106
Add terms
Multiply
Calculate quotient
At the beginning of the lesson, Diego's father took out a loan from the National Bank to start his own company. He will start his repayment schedule by paying $2400 in the first month, after which the monthly amount will decrease by $100 each month. In the second month he will pay $2300, in the third month $2200, and so on.
a1=2400, d=-100
Distribute -100
Add terms
n=5
(-a)b=-ab
Add terms