Question 1

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so.
-3, 6, 12, 24, 48

Accepted Answer

The common ratio is found by dividing any term by its preceding term, e.g., 6/3 = 2, 12/6 = 2, etc., showing a consistent common ratio of 2.

Question 2

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so.
-29, 17.4, 10.44, 6.264, 3.76

Accepted Answer

The common ratio of a geometric sequence is found by dividing any term by the previous term. For this sequence, 17.4 / 29 = 0.6, 10.44 / 17.4 = 0.6, and so on. This confirms the sequence is geometric with a common ratio of 0.6.

Question 3

Solve the problem.
-Suppose you just received a job offer with a starting salary of $37,000 per year and a guaranteed raise of $1500
per year. How many years will it be before you've made a total (or aggregate) salary of $1,025,000?

Accepted Answer

The answer of Solve the problem.
-Suppose you just received a...

Question 4

Determine whether the sequence is geometric.
-2, 6, 18, 54, 162, ...

Accepted Answer

The sequence is geometric because each term is obtained by multiplying the previous term by a common ratio, which is 3 in this case (6/2 = 18/6 = 54/18 = 162/54 = 3).

Question 5

Determine whether the sequence is geometric.
-4, 12, 36, 108, 324, ...

Accepted Answer

Each term is obtained by multiplying the previous term by 3, which is a constant ratio, making it a geometric sequence.

Question 6

Solve the problem.
-A brick staircase has a total of 16 steps The bottom step requires 117 bricks. Each successive step requires 5 less bricks than the prior one. How many bricks are required to build the staircase?

Accepted Answer

The number of bricks required for each step forms an arithmetic sequence where the first term (a1) is 117 and the common difference (d) is -5. The number of bricks for the nth step is given by the formula: a_n = a1 + (n-1)d. The total number of bricks is the sum of the sequence: S_n = n/2 * (a1 + a_n). For 16 steps, a_16 = 117 + 15*(-5) = 42. The sum is S_16 = 16/2 * (117 + 42) = 1272 bricks.

Question 7

Solve the problem.
-A local civic theater has 22 seats in the first row and 21 rows in all. Each successive row contains 3 additional seats. How many seats are in the civic theater?

Accepted Answer

This is an arithmetic series problem where the first term (a) is 22, the number of terms (n) is 21, and the common difference (d) is 3. The formula for the sum of an arithmetic series is S = n/2[2a + (n-1)d]. Plugging in the values gives S = 21/2[2(22) + (21-1)3] = 21/2[44 + 60] = 21/2[104] = 1092.

Question 8

Find the sum of the arithmetic sequence.
-{-5n + 5}, n = 29

Accepted Answer

The answer of Find the sum of the arithmetic sequence.
-{-5n...

Question 9

Use a graphing utility to find the sum of each sequence.
-{ 3.6n + 8.83}, n=10

Accepted Answer

The answer of Use a graphing utility to find the...

Question 10

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so.
-1, -3, 9, -27, 81

Accepted Answer

The common ratio is -3, as each term is obtained by multiplying the previous term by -3 (e.g., 1 * -3 = -3, -3 * -3 = 9, etc.).

Question 11

Determine whether the sequence is geometric.
-3, 1, -1, -3, -5, ...

Accepted Answer

A geometric sequence requires that each term be obtained by multiplying the previous term by a constant called the common ratio. In this sequence, the pattern involves adding a constant difference (-2) rather than multiplying by a constant ratio, making it an arithmetic sequence, not geometric.

Question 12

Determine whether the sequence is geometric.
-3, 5, 7, 11, 13, ...

Accepted Answer

A geometric sequence requires each term to be obtained by multiplying the previous term by a constant called the common ratio. The given sequence does not have a constant ratio between consecutive terms, so it is not geometric.

Question 13

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so.
-$$\frac { 2 } { 3 } , \frac { 8 } { 3 } , \frac { 32 } { 3 } , \frac { 128 } { 3 } , \frac { 512 } { 3 }$$

Accepted Answer

The sequence is geometric because each term after the first is obtained by multiplying the previous term by a constant. The common ratio can be found by dividing any term by its preceding term, for example, $\frac{8}{3} ÷ \frac{2}{3} = 4$ or $\frac{32}{3} ÷ \frac{8}{3} = 4$. Thus, the common ratio is 4.

Question 14

Determine whether the sequence is geometric.
-4, -12, 36, -108, 324, ...

Accepted Answer

The sequence is geometric with a common ratio of -3.

Question 15

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so.
-$\frac { 1 } { 9 } , \frac { 1 } { 14 } , \frac { 1 } { 19 } , \frac { 1 } { 24 }$

Accepted Answer

The sequence is not geometric because the ratio between consecutive terms is not constant. Each term is obtained by subtracting 5 from the denominator of the previous term, which does not result in a constant multiplication factor.

Question 16

Solve the problem.
-A theater has 34 rows with 24 seats in the first row, 29 in the second row, 34 in the third row, and so forth. How many seats are in the theater?

Accepted Answer

The answer of Solve the problem.
-A theater has 34 rows...

Question 17

Find the sum of the arithmetic sequence.
-{-3n - 1}, n = 50

Accepted Answer

To find the sum of the arithmetic sequence {-3n - 1} for n = 50, we first need to determine the first term (a1) and the 50th term (a50) of the sequence, and then use the formula for the sum of an arithmetic sequence: S = n/2 * (a1 + a50), where n is the number of terms.The first term (a1) is when n = 1:a1 = -3(1) - 1 = -3 - 1 = -4The 50th term (a50) is when n = 50:a50 = -3(50) - 1 = -150 - 1 = -151Now, using the sum formula:S = 50/2 * (-4 - 151) = 25 * (-155) = -3875

Question 18

Determine whether the sequence is geometric.
-$$\frac { 1 } { 8 } , \frac { 1 } { 11 } , \frac { 1 } { 14 } , \frac { 1 } { 17 } , \ldots$$

Accepted Answer

A sequence is geometric if each term after the first is obtained by multiplying the preceding term by a fixed, non-zero number called the common ratio. In this sequence, the denominator increases by 3 each time, which does not correspond to multiplication by a constant factor. Therefore, the sequence is not geometric.

Question 19

Determine whether the sequence is geometric.
-7, 4, 1, -2, -5, ...

Accepted Answer

The sequence is not geometric because the ratio between consecutive terms is not constant.

Question 20

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so.
-$$\frac { 3 } { 1 } , \frac { 3 } { 4 } , \frac { 3 } { 16 } , \frac { 3 } { 64 } , \frac { 3 } { 256 }$$

Accepted Answer

The common ratio can be found by dividing any term in the sequence by the term preceding it. For example, $\frac{3/4}{3/1} = \frac{1}{4}$ or $\frac{3/16}{3/4} = \frac{1}{4}$. Thus, the common ratio is $\frac{1}{4}$.

Quiz 7: Sequences; Induction; the Binomial Theorem

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so. -3, 6, 12, 24, 48

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so. -29, 17.4, 10.44, 6.264, 3.76

Solve the problem. -Suppose you just received a job offer with a starting salary of $37,000 per year and a guaranteed raise of $1500 per year. How many years will it be before you've made a total (or aggregate) salary of $1,025,000?

Determine whether the sequence is geometric. -2, 6, 18, 54, 162, ...

Determine whether the sequence is geometric. -4, 12, 36, 108, 324, ...

Solve the problem. -A brick staircase has a total of 16 steps The bottom step requires 117 bricks. Each successive step requires 5 less bricks than the prior one. How many bricks are required to build the staircase?

Solve the problem. -A local civic theater has 22 seats in the first row and 21 rows in all. Each successive row contains 3 additional seats. How many seats are in the civic theater?

Find the sum of the arithmetic sequence. -{-5n + 5}, n = 29

Use a graphing utility to find the sum of each sequence. -{ 3.6n + 8.83}, n=10

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so. -1, -3, 9, -27, 81

Determine whether the sequence is geometric. -3, 1, -1, -3, -5, ...

Determine whether the sequence is geometric. -3, 5, 7, 11, 13, ...

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so. - $\frac { 2 } { 3 } , \frac { 8 } { 3 } , \frac { 32 } { 3 } , \frac { 128 } { 3 } , \frac { 512 } { 3 }$

Determine whether the sequence is geometric. -4, -12, 36, -108, 324, ...

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so. - $\frac { 1 } { 9 } , \frac { 1 } { 14 } , \frac { 1 } { 19 } , \frac { 1 } { 24 }$

Solve the problem. -A theater has 34 rows with 24 seats in the first row, 29 in the second row, 34 in the third row, and so forth. How many seats are in the theater?

Find the sum of the arithmetic sequence. -{-3n - 1}, n = 50

Determine whether the sequence is geometric. - $\frac { 1 } { 8 } , \frac { 1 } { 11 } , \frac { 1 } { 14 } , \frac { 1 } { 17 } , \ldots$

Determine whether the sequence is geometric. -7, 4, 1, -2, -5, ...

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so. - $\frac { 3 } { 1 } , \frac { 3 } { 4 } , \frac { 3 } { 16 } , \frac { 3 } { 64 } , \frac { 3 } { 256 }$