Take Note Key Concept: Rule For an Arithmetic Sequence

The nth term of an arithmetic sequence with first term A(1) and common difference d is given by

n th term first term term number ↓ ↓ ↓ A ( n ) = A ( 1 ) + ( n − 1 ) d ↗ common difference table with 5 rows and 1 column , row1 column 1 , n , thterm . firstterm . termnumber , row2 column 1 , downwards arrow downwards arrow downwards arrow , row3 column 1 , eh open n close equals eh open 1 close plus open n minus 1 close d , row4 column 1 , north east arrow , row5 column 1 , commondifference , end table

Problem 3 Writing a Rule for an Arithmetic Sequence

Online Auction An online auction works as shown below. Write a rule to represent the bids as an arithmetic sequence. What is the twelfth bid?

Make a table of the bids. Identify the first term and common difference.

Image Long Description

Substitute A ( 1 ) = 200 eh open 1 close equals 200  and d = 10 d equals 10  into the formula A ( n ) = A ( 1 ) + ( n − 1 ) d . eh open n close equals eh open 1 close plus open n minus 1 close d .  The rule A ( n ) = 200 + ( n − 1 ) 10 eh open n close equals 200 plus open n minus 1 close 10  represents the sequence of the auction bids. To find the twelfth bid, evaluate A(n) for n = 12 . n equals 12 .

A ( 12 ) = 200 + ( 12 − 1 ) 10 = 310 eh open 12 close equals 200 plus open 12 minus 1 close 10 equals 310

The twelfth bid is $310.

✓ Got It?

3. 1. A subway pass has a starting value of $100. After one ride, the value of the pass is $98.25. After two rides, its value is $96.50. After three rides, its value is $94.75. Write a rule to represent the remaining value on the card as an arithmetic sequence. What is the value of the pass after 15 rides?
   2. Reasoning How many rides can be taken with the $100 pass?