Sequences and Series: Arithmetic and Geometric
Finding patterns, rules, terms, and sums
Sequences and Series: Arithmetic and Geometric
Finding patterns, rules, terms, and sums
Math - Grade 9-12
- 1
Determine whether the sequence 5, 8, 11, 14, ... is arithmetic or geometric. Then state the common difference or common ratio.
Check the change from one term to the next.
This sequence is arithmetic because each term increases by 3. The common difference is 3. - 2
Determine whether the sequence 2, 6, 18, 54, ... is arithmetic or geometric. Then state the common difference or common ratio.
Compare each term to the one before it using division.
This sequence is geometric because each term is multiplied by 3. The common ratio is 3. - 3
Find the 10th term of the arithmetic sequence 4, 9, 14, 19, ... .
The 10th term is 49. The sequence has first term 4 and common difference 5, so a10 = 4 + 9(5) = 49. - 4
Find the 7th term of the geometric sequence 3, 6, 12, 24, ... .
Use the formula for a geometric sequence: an = a1(r^(n-1)).
The 7th term is 192. The sequence has first term 3 and common ratio 2, so a7 = 3(2^6) = 192. - 5
Write an explicit formula for the arithmetic sequence with first term 12 and common difference -4.
An explicit formula is an = 12 + (n - 1)(-4), which can also be written as an = 16 - 4n. - 6
Write an explicit formula for the geometric sequence with first term 7 and common ratio 2.
A geometric sequence multiplies by the same number each time.
An explicit formula is an = 7(2^(n - 1)). - 7
Find the sum of the first 12 terms of the arithmetic series 3 + 7 + 11 + 15 + ... .
The sum of the first 12 terms is 300. Here a1 = 3, d = 4, and a12 = 47, so S12 = 12(3 + 47) / 2 = 300. - 8
Find the sum of the first 6 terms of the geometric series 5 + 10 + 20 + 40 + ... .
Use the finite geometric series formula.
The sum of the first 6 terms is 315. Using a1 = 5 and r = 2, S6 = 5(2^6 - 1) / (2 - 1) = 315. - 9
The 4th term of an arithmetic sequence is 17 and the 11th term is 45. Find the first term and the common difference.
The first term is 5 and the common difference is 4. Since a4 = a1 + 3d = 17 and a11 = a1 + 10d = 45, subtracting gives 7d = 28, so d = 4, and then a1 = 5. - 10
The 2nd term of a geometric sequence is 12 and the 5th term is 324. Find the first term and the common ratio.
Divide the later term equation by the earlier one to solve for the ratio.
The first term is 4 and the common ratio is 3. Since a2 = a1r = 12 and a5 = a1r^4 = 324, dividing gives r^3 = 27, so r = 3, and then a1 = 4. - 11
Find the next three terms of the arithmetic sequence 22, 17, 12, 7, ... .
The next three terms are 2, -3, and -8. The sequence decreases by 5 each time. - 12
Find the next three terms of the geometric sequence 160, 80, 40, 20, ... .
Look for a constant ratio between consecutive terms.
The next three terms are 10, 5, and 2.5. The sequence is multiplied by 1/2 each time. - 13
Find the 15th term of the arithmetic sequence with first term -6 and common difference 7.
The 15th term is 92. Using an = a1 + (n - 1)d, we get a15 = -6 + 14(7) = 92. - 14
Find the 8th term of the geometric sequence with first term 256 and common ratio 1/2.
Multiply by 1/2 repeatedly or use exponents.
The 8th term is 2. Using an = a1(r^(n - 1)), we get a8 = 256(1/2)^7 = 2. - 15
A theater has 18 seats in the first row and each row behind it has 3 more seats than the row before. How many seats are in the 20th row, and how many seats are in the first 20 rows altogether?
There are 75 seats in the 20th row and 930 seats in the first 20 rows altogether. This is an arithmetic sequence with a1 = 18 and d = 3, so a20 = 18 + 19(3) = 75 and S20 = 20(18 + 75) / 2 = 930.