12.2. Arithmetic and Geometric Sequences http://www.ck12.org
different way.
− 3 , 2 , 7 , 12 , 17 ,...
You should see the common difference of 5. This means the common difference from the original sequence is 5
3. The equation isan=−^1 +(n−^1 )
( 5
3). The 10thterm is:− 1 + 9 ·(^53 )=− 1 + 3 · 5 =− 1 + 15 = 14
- The sequence is not arithmetic nor geometric. It will help to find the pattern by examining the common differences
and then the common differences of the common differences. This numerical process is connected to ideas in
calculus.
0, 3, 8, 15, 24, 35
3, 5, 7, 9, 11
2, 2, 2, 2
Notice when you examine the common difference of the common differences the pattern becomes increasingly
clear. Since it tooktwolayers to find a constant function, this pattern isquadraticand very similar to the perfect
squares.
1 , 4 , 9 , 16 , 25 , 36 ,...
Theakterm can be described asak=k^2 − 1
Practice
Use the sequence 1, 5 , 9 , 13 ,...for questions 1-3.
- Find the next three terms in the sequence.
- Find an equation that defines theakterm of the sequence.
- Find the 150thterm of the sequence.
Use the sequence 12, 4 ,^43 ,^49 ,...for questions 4-6. - Find the next three terms in the sequence.
- Find an equation that defines theakterm of the sequence.
- Find the 17thterm of the sequence.
Use the sequence 10,− 2 ,^25 ,− 252 ,...for questions 7-9. - Find the next three terms in the sequence.
- Find an equation that defines theakterm of the sequence.
- Find the 12thterm of the sequence.
Use the sequence^72 ,^92 ,^112 ,^132 ,...for questions 10-12. - Find the next three terms in the sequence.
- Find an equation that defines theakterm of the sequence.
- Find the 314thterm of the sequence.
- Find an equation that defines theakterm for the sequence 4, 11 , 30 , 67 ,...
- Explain the connections between arithmetic sequences and linear functions.
- Explain the connections between geometric sequences and exponential functions.