Question 1

Write out the first three terms and the last term of the arithmetic sequence.
-$$\sum _ { i = 1 } ^ { 70 } - 2 i$$

Accepted Answer

The given sequence is an arithmetic sequence where each term is obtained by subtracting 2 times the position of the term. The first term is $-2 \times 1 = -2$, the second term is $-2 \times 2 = -4$, and the third term is $-2 \times 3 = -6$. The last term, when $i = 70$, is $-2 \times 70 = -140$. This matches option D.

Question 2

Solve the problem.
-The population of a town is increasing by 300 inhabitants each year. If its current population is 20,506 and this trend continues, what would its population be in 8 years?

Accepted Answer

The population increases by 300 inhabitants each year. In 8 years, the increase will be 300 inhabitants/year * 8 years = 2,400 inhabitants. Adding this to the current population of 20,506 gives 20,506 + 2,400 = 22,906 inhabitants.

Question 3

Use the partial sum formula to find the partial sum of the given arithmetic sequence.
-Find the sum of the first eight terms of the arithmetic sequence: 3, 5, 7, . . . .

Accepted Answer

The answer of Use the partial sum formula to find...

Question 4

Write out the first three terms and the last term of the arithmetic sequence.
-$$\sum _ { i = 1 } ^ { 12 }$$

Accepted Answer

The correct sequence is $0 + 5 + 10 + \ldots + 55$, which starts at 0 and increases by 5 each time, making it an arithmetic sequence with the first term 0, the common difference 5, and the last term 55.

Question 5

Use the partial sum formula to find the partial sum of the given arithmetic sequence.
-Find the sum of the first 25 terms of the arithmetic sequence: -5, 2, 9, 16, . . . .

Accepted Answer

The answer of Use the partial sum formula to find...

Question 6

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a₁₄ when a₁ = 23, d = -3.

Accepted Answer

The answer of Use the formula for the general term...

Question 7

Solve the problem.
-To train for a race, Will begins by jogging 11 minutes one day per week. He increases his jogging time by 6 minutes each week. Write the general term of this arithmetic sequence, and find how many weeks it takes for him to reach a jogging time of one hour.

Accepted Answer

The answer of Solve the problem.
-To train for a race,...

Question 8

Use the partial sum formula to find the partial sum of the given arithmetic sequence.
-Find the sum of the odd integers between 22 and 52.

Accepted Answer

The answer of Use the partial sum formula to find...

Question 9

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
-$$\sum _ { i = 1 } ^ { 26 } ( - 3 i + 8 )$$

Accepted Answer

The sum of an arithmetic sequence can be found using the formula $S_n = \frac{n}{2} [2a + (n - 1)d]$, where $n$ is the number of terms, $a$ is the first term, and $d$ is the common difference. Here, $a = -3(1) + 8 = 5$ and $d = -3$. Plugging these values into the formula with $n = 26$, we get $S_{26} = \frac{26}{2} [2(5) + (26 - 1)(-3)] = 13[10 - 75] = 13(-65) = -845$.

Question 10

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a ₃₇ when $a _ { 1 } = 5 , d = \frac { 2 } { 3 }$.

Accepted Answer

The answer of Use the formula for the general term...

Question 11

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a ₃₂ when a1 = 3 , d = -4 .

Accepted Answer

The formula for the nth term of an arithmetic sequence is a_n = a1 + (n-1)d. Plugging in the values: a_32 = 3 + (32-1)(-4) = 3 + 31(-4) = 3 - 124 = -121.

Question 12

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
-$$\sum _ { \mathrm { i } = 1 } ^ { 49 } ( 5 \mathrm { i } + 5 )$$

Accepted Answer

The answer of Use the formula for the sum of...

Question 13

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a8 when a₁ = -9 , d = 4 .

Accepted Answer

The formula for the nth term of an arithmetic sequence is a_n = a_1 + (n-1)d. Substituting the given values: a_8 = -9 + (8-1) * 4 = -9 + 28 = 19.

Question 14

Use the partial sum formula to find the partial sum of the given arithmetic sequence.
-Find 2 + 4 + 6 + 8 + . . ., the sum of the first 40 positive even integers.

Accepted Answer

The sequence is an arithmetic sequence with the first term a = 2, the common difference d = 2, and the number of terms n = 40. The formula for the sum of the first n terms of an arithmetic sequence is S_n = n/2 * (2a + (n-1)d). Plugging in the values, S_40 = 40/2 * (2*2 + (40-1)*2) = 20 * (4 + 78) = 20 * 82 = 1640.

Question 15

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d.
-$$a _ { 1 } = - \frac { 3 } { 5 } , d = \frac { 4 } { 5 }$$

Accepted Answer

The formula for the nth term of an arithmetic sequence is $a_n = a_1 + (n - 1)d$. Substituting $a_1 = -\frac{3}{5}$ and $d = \frac{4}{5}$, we get $a_n = -\frac{3}{5} + (n - 1)\frac{4}{5} = \frac{4}{5}n - \frac{7}{5}$. For $n = 20$, $a_{20} = \frac{4}{5} \times 20 - \frac{7}{5} = \frac{73}{5}$.

Question 16

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d.
-$$a _ { 1 } = - 9 , d = - 0.5$$

Accepted Answer

The formula for the nth term of an arithmetic sequence is $a_n = a_1 + (n - 1)d$. Substituting $a_1 = -9$ and $d = -0.5$ gives $a_n = -9 + (n - 1)(-0.5) = -9 - 0.5n + 0.5 = -0.5n - 8.5$. For $n = 20$, $a_{20} = -0.5(20) - 8.5 = -10 - 8.5 = -18.5$.

Question 17

Use the partial sum formula to find the partial sum of the given arithmetic sequence.
-Find the sum of the first five terms of the arithmetic sequence: -10, -14, -18, . . . .

Accepted Answer

The first term (a1) is -10, the common difference (d) is -4, and the number of terms (n) is 5. The formula for the sum of an arithmetic sequence is S_n = n/2 * (2a1 + (n-1)d). Plugging in the values: S_5 = 5/2 * (2(-10) + 4(-4)) = 5/2 * (-20 - 16) = 5/2 * (-36) = -90.

Question 18

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
-$$\sum _ { i = 1 } ^ { 27 } ( 6 i - 1 )$$

Accepted Answer

The answer of Use the formula for the sum of...

Question 19

Solve the problem.
-Jacie is considering a job that offers a monthly starting salary of $3500 and guarantees her a monthly raise of $110 during her first year on the job. Find the general term of this arithmetic sequence and her monthly salary at the end of her first year.

Accepted Answer

The general term of an arithmetic sequence is given by $a_n = a_1 + (n - 1)d$, where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number. Here, $a_1 = 3500$ and $d = 110$, but the formula in choice A mistakenly uses 3390 as the starting value. However, to calculate the salary at the end of the first year (12th month), we use $a_{12} = 3500 + (12 - 1) \times 110 = 3500 + 11 \times 110 = 4710$, which matches the outcome in A despite the initial term discrepancy in the formula. The correct formula should be $a_n = 3500 + 110(n - 1)$, but due to the options provided and the correct final salary, A is chosen based on the end salary match. My explanation contains an error in selecting the correct choice based on the initial premise; the correct formula and end salary align with option C, not A, as initially stated.

Question 20

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a₅₀ when a₁ = -9, d = 3.

Accepted Answer

The formula for the nth term of an arithmetic sequence is a_n = a_1 + (n-1)d. Plugging in the values: a_50 = -9 + (50-1) * 3 = -9 + 147 = 138.

Quiz 8: Quadratic Equations and Functions

Write out the first three terms and the last term of the arithmetic sequence. - $\sum _ { i = 1 } ^ { 70 } - 2 i$

Solve the problem. -The population of a town is increasing by 300 inhabitants each year. If its current population is 20,506 and this trend continues, what would its population be in 8 years?

Use the partial sum formula to find the partial sum of the given arithmetic sequence. -Find the sum of the first eight terms of the arithmetic sequence: 3, 5, 7, . . . .

Write out the first three terms and the last term of the arithmetic sequence. - $\sum _ { i = 1 } ^ { 12 }$

Use the partial sum formula to find the partial sum of the given arithmetic sequence. -Find the sum of the first 25 terms of the arithmetic sequence: -5, 2, 9, 16, . . . .

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a₁₄ when a₁ = 23, d = -3.

Solve the problem. -To train for a race, Will begins by jogging 11 minutes one day per week. He increases his jogging time by 6 minutes each week. Write the general term of this arithmetic sequence, and find how many weeks it takes for him to reach a jogging time of one hour.

Use the partial sum formula to find the partial sum of the given arithmetic sequence. -Find the sum of the odd integers between 22 and 52.

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - $\sum _ { i = 1 } ^ { 26 } ( - 3 i + 8 )$

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a ₃₇ when $a _ { 1 } = 5 , d = \frac { 2 } { 3 }$ .

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a ₃₂ when a1 = 3 , d = -4 .

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - $\sum _ { \mathrm { i } = 1 } ^ { 49 } ( 5 \mathrm { i } + 5 )$

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a8 when a₁ = -9 , d = 4 .

Use the partial sum formula to find the partial sum of the given arithmetic sequence. -Find 2 + 4 + 6 + 8 + . . ., the sum of the first 40 positive even integers.

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. - $a _ { 1 } = - \frac { 3 } { 5 } , d = \frac { 4 } { 5 }$

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. - $a _ { 1 } = - 9 , d = - 0.5$

Use the partial sum formula to find the partial sum of the given arithmetic sequence. -Find the sum of the first five terms of the arithmetic sequence: -10, -14, -18, . . . .

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - $\sum _ { i = 1 } ^ { 27 } ( 6 i - 1 )$

Solve the problem. -Jacie is considering a job that offers a monthly starting salary of $3500 and guarantees her a monthly raise of $110 during her first year on the job. Find the general term of this arithmetic sequence and her monthly salary at the end of her first year.

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a₅₀ when a₁ = -9, d = 3.

Quiz 8: Quadratic Equations and Functions

Write out the first three terms and the last term of the arithmetic sequence. - ∑i=170−2i\sum _ { i = 1 } ^ { 70 } - 2 i∑i=170​−2i

Solve the problem. -The population of a town is increasing by 300 inhabitants each year. If its current population is 20,506 and this trend continues, what would its population be in 8 years?

Use the partial sum formula to find the partial sum of the given arithmetic sequence. -Find the sum of the first eight terms of the arithmetic sequence: 3, 5, 7, . . . .

Write out the first three terms and the last term of the arithmetic sequence. - ∑i=112\sum _ { i = 1 } ^ { 12 }∑i=112​

Use the partial sum formula to find the partial sum of the given arithmetic sequence. -Find the sum of the first 25 terms of the arithmetic sequence: -5, 2, 9, 16, . . . .

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a14 when a1 = 23, d = -3.

Solve the problem. -To train for a race, Will begins by jogging 11 minutes one day per week. He increases his jogging time by 6 minutes each week. Write the general term of this arithmetic sequence, and find how many weeks it takes for him to reach a jogging time of one hour.

Use the partial sum formula to find the partial sum of the given arithmetic sequence. -Find the sum of the odd integers between 22 and 52.

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - ∑i=126(−3i+8) \sum _ { i = 1 } ^ { 26 } ( - 3 i + 8 ) ∑i=126​(−3i+8)

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a 37 when a1=5,d=23a _ { 1 } = 5 , d = \frac { 2 } { 3 }a1​=5,d=32​ .

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a 32 when a1 = 3 , d = -4 .

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - ∑i=149(5i+5) \sum _ { \mathrm { i } = 1 } ^ { 49 } ( 5 \mathrm { i } + 5 ) ∑i=149​(5i+5)

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a8 when a1 = -9 , d = 4 .

Use the partial sum formula to find the partial sum of the given arithmetic sequence. -Find 2 + 4 + 6 + 8 + . . ., the sum of the first 40 positive even integers.

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. - a1=−35,d=45a _ { 1 } = - \frac { 3 } { 5 } , d = \frac { 4 } { 5 }a1​=−53​,d=54​

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. - a1=−9,d=−0.5a _ { 1 } = - 9 , d = - 0.5a1​=−9,d=−0.5

Use the partial sum formula to find the partial sum of the given arithmetic sequence. -Find the sum of the first five terms of the arithmetic sequence: -10, -14, -18, . . . .

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - ∑i=127(6i−1) \sum _ { i = 1 } ^ { 27 } ( 6 i - 1 ) ∑i=127​(6i−1)

Solve the problem. -Jacie is considering a job that offers a monthly starting salary of $3500 and guarantees her a monthly raise of $110 during her first year on the job. Find the general term of this arithmetic sequence and her monthly salary at the end of her first year.

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a50 when a1 = -9, d = 3.

Write out the first three terms and the last term of the arithmetic sequence. - $\sum _ { i = 1 } ^ { 70 } - 2 i$

Write out the first three terms and the last term of the arithmetic sequence. - $\sum _ { i = 1 } ^ { 12 }$

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a₁₄ when a₁ = 23, d = -3.

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - $\sum _ { i = 1 } ^ { 26 } ( - 3 i + 8 )$

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a ₃₇ when $a _ { 1 } = 5 , d = \frac { 2 } { 3 }$ .

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a ₃₂ when a1 = 3 , d = -4 .

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - $\sum _ { \mathrm { i } = 1 } ^ { 49 } ( 5 \mathrm { i } + 5 )$

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a8 when a₁ = -9 , d = 4 .

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. - $a _ { 1 } = - \frac { 3 } { 5 } , d = \frac { 4 } { 5 }$

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. - $a _ { 1 } = - 9 , d = - 0.5$

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - $\sum _ { i = 1 } ^ { 27 } ( 6 i - 1 )$

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d. -Find a₅₀ when a₁ = -9, d = 3.