Sequences and Series: Arithmetic and Geometric Worksheet

Question 1

Determine whether the sequence 5, 8, 11, 14, ... is arithmetic or geometric. Then state the common difference or common ratio.

Accepted Answer

This sequence is arithmetic because each term increases by 3. The common difference is 3.

Question 2

Determine whether the sequence 2, 6, 18, 54, ... is arithmetic or geometric. Then state the common difference or common ratio.

Accepted Answer

This sequence is geometric because each term is multiplied by 3. The common ratio is 3.

Question 3

Find the 10th term of the arithmetic sequence 4, 9, 14, 19, ... .

Accepted Answer

The 10th term is 49. The sequence has first term 4 and common difference 5, so a10 = 4 + 9(5) = 49.

Question 4

Find the 7th term of the geometric sequence 3, 6, 12, 24, ... .

Accepted Answer

The 7th term is 192. The sequence has first term 3 and common ratio 2, so a7 = 3(2^6) = 192.

Question 5

Write an explicit formula for the arithmetic sequence with first term 12 and common difference -4.

Accepted Answer

An explicit formula is an = 12 + (n - 1)(-4), which can also be written as an = 16 - 4n.

Question 6

Write an explicit formula for the geometric sequence with first term 7 and common ratio 2.

Accepted Answer

An explicit formula is an = 7(2^(n - 1)).

Question 7

Find the sum of the first 12 terms of the arithmetic series 3 + 7 + 11 + 15 + ... .

Accepted Answer

The sum of the first 12 terms is 300. Here a1 = 3, d = 4, and a12 = 47, so S12 = 12(3 + 47) / 2 = 300.

Question 8

Find the sum of the first 6 terms of the geometric series 5 + 10 + 20 + 40 + ... .

Accepted Answer

The sum of the first 6 terms is 315. Using a1 = 5 and r = 2, S6 = 5(2^6 - 1) / (2 - 1) = 315.

Question 9

The 4th term of an arithmetic sequence is 17 and the 11th term is 45. Find the first term and the common difference.

Accepted Answer

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.

Question 10

The 2nd term of a geometric sequence is 12 and the 5th term is 324. Find the first term and the common ratio.

Accepted Answer

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.

Question 11

Find the next three terms of the arithmetic sequence 22, 17, 12, 7, ... .

Accepted Answer

The next three terms are 2, -3, and -8. The sequence decreases by 5 each time.

Question 12

Find the next three terms of the geometric sequence 160, 80, 40, 20, ... .

Accepted Answer

The next three terms are 10, 5, and 2.5. The sequence is multiplied by 1/2 each time.

Question 13

Find the 15th term of the arithmetic sequence with first term -6 and common difference 7.

Accepted Answer

The 15th term is 92. Using an = a1 + (n - 1)d, we get a15 = -6 + 14(7) = 92.

Question 14

Find the 8th term of the geometric sequence with first term 256 and common ratio 1/2.

Accepted Answer

The 8th term is 2. Using an = a1(r^(n - 1)), we get a8 = 256(1/2)^7 = 2.

Question 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?

Accepted Answer

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.

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Sequences and Series: Arithmetic and Geometric

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