Precalculus: Arithmetic and Geometric Sequences
Finding terms, formulas, and sums for common sequence patterns
Precalculus: Arithmetic and Geometric Sequences
Finding terms, formulas, and sums for common sequence patterns
Math - Grade 9-12
- 1
The sequence 7, 12, 17, 22, ... is arithmetic. Find the common difference and write an explicit formula for a_n.
Subtract consecutive terms to find the common difference.
The common difference is 5. An explicit formula is a_n = 7 + 5(n - 1), which can also be written as a_n = 5n + 2. - 2
The sequence 3, 6, 12, 24, ... is geometric. Find the common ratio and write an explicit formula for a_n.
Divide a term by the previous term to find the common ratio.
The common ratio is 2. An explicit formula is a_n = 3(2)^(n - 1). - 3
Find the 25th term of the arithmetic sequence with a_1 = -4 and common difference d = 6.
The 25th term is 140 because a_25 = -4 + 6(25 - 1) = -4 + 144 = 140. - 4
Find the 8th term of the geometric sequence with a_1 = 5 and common ratio r = -2.
Use a_n = a_1r^(n - 1).
The 8th term is -640 because a_8 = 5(-2)^(8 - 1) = 5(-128) = -640. - 5
Determine whether the sequence 81, 27, 9, 3, 1, ... is arithmetic, geometric, or neither. Explain your reasoning.
The sequence is geometric because each term is found by multiplying the previous term by 1/3. The common ratio is 1/3. - 6
Determine whether the sequence 4, 9, 16, 25, 36, ... is arithmetic, geometric, or neither. Explain your reasoning.
Check both consecutive differences and consecutive ratios.
The sequence is neither arithmetic nor geometric. The differences are 5, 7, 9, and 11, so there is no common difference, and the ratios are not constant. - 7
Write a recursive formula for the arithmetic sequence 15, 11, 7, 3, ...
A recursive formula is a_1 = 15 and a_n = a_(n - 1) - 4 for n >= 2. The sequence decreases by 4 each term. - 8
Write a recursive formula for the geometric sequence 2, -6, 18, -54, ...
Find the number you multiply by to get from one term to the next.
A recursive formula is a_1 = 2 and a_n = -3a_(n - 1) for n >= 2. Each term is multiplied by -3. - 9
An arithmetic sequence has a_4 = 18 and a_10 = 42. Find a_1 and write an explicit formula for a_n.
Use the two known terms to find how much the sequence changes over 6 steps.
The common difference is 4 because (42 - 18) divided by (10 - 4) equals 4. Since a_4 = a_1 + 3d, 18 = a_1 + 12, so a_1 = 6. An explicit formula is a_n = 6 + 4(n - 1), or a_n = 4n + 2. - 10
A geometric sequence has a_2 = 12 and a_5 = 324. If the common ratio is positive, find a_1 and write an explicit formula for a_n.
From a_5 = a_2r^3, 324 = 12r^3, so r^3 = 27 and r = 3. Since a_2 = a_1r, 12 = 3a_1, so a_1 = 4. An explicit formula is a_n = 4(3)^(n - 1). - 11
Find the sum of the first 20 terms of the arithmetic sequence 8, 13, 18, 23, ...
Use S_n = n(a_1 + a_n)/2.
The sum of the first 20 terms is 1110. The 20th term is 8 + 5(19) = 103, so S_20 = 20(8 + 103)/2 = 1110. - 12
Find the sum of the first 7 terms of the geometric sequence 6, 18, 54, 162, ...
Use the finite geometric series formula with a_1 = 6, r = 3, and n = 7.
The sum of the first 7 terms is 6558. Using S_n = a_1(1 - r^n)/(1 - r), S_7 = 6(1 - 3^7)/(1 - 3) = 6558. - 13
A theater has 18 seats in the first row, 22 seats in the second row, 26 seats in the third row, and so on. If there are 30 rows, how many seats are in the theater?
There are 2280 seats in the theater. The rows form an arithmetic sequence with a_1 = 18 and d = 4. The 30th row has 18 + 4(29) = 134 seats, so the total is S_30 = 30(18 + 134)/2 = 2280. - 14
A population of bacteria starts at 500 and triples every hour. Write a formula for the population after n hours, where n = 0 represents the starting time, and find the population after 6 hours.
Because n = 0 is the starting time, the exponent is n, not n - 1.
A formula is P(n) = 500(3)^n because the starting amount is 500 and the population triples each hour. After 6 hours, P(6) = 500(3^6) = 364500 bacteria. - 15
A student saves $40 in week 1, $55 in week 2, $70 in week 3, and continues this pattern. How much will the student save in week 12, and what is the total amount saved over the 12 weeks?
This is an arithmetic sequence with common difference 15.
The student will save $205 in week 12 because a_12 = 40 + 15(11) = 205. The total saved over 12 weeks is $1470 because S_12 = 12(40 + 205)/2 = 1470.