Question 1

Assume that the sequence is defined recursively.Find the first three terms of the sequence.​ $$a _ { 1 } = 3 \text { and } a _ { n + 1 } = 4 a _ { n } ^ { 2 }$$ ​
A)36
B)5184
C)3
D)4
E)32

Accepted Answer

The answer of Assume that the sequence is defined recursively.Find...

Question 2

A deposit of $ 5000 is made in an account that earns 8 % interest compounded monthly.The balance in the account after n months is given by $$A _ { n } = 5000 \left( 1 + \frac { 0.08 } { 12 } \right) ^ { n } , n = 1,2,3 , \ldots$$ Find the balance in the account after 8 years by finding the 96th term of the sequence.Round to the nearest penny.
A)$ 8,084,450.96
B)$ 9462.29
C)$ 9399.62
D)$ 483200.00
E)$ 9525.37

Accepted Answer

To find the balance after 8 years (which is 96 months since 8 years * 12 months/year = 96 months), we substitute $n = 96$ into the given formula: $$A_{96} = 5000 \left(1 + \frac{0.08}{12}\right)^{96}$$. Calculating this gives approximately $9462.29.

Question 3

Use sigma notation to write the sum. $$\frac { 1 } { 3 \cdot 2 } + \frac { 1 } { 4 \cdot 3 } + \ldots + \frac { 1 } { 8 \cdot 7 }$$
A) $\sum _ { n = 1 } ^ { 6 } \frac { 1 } { ( n + 1 ) ( n + 2 ) }$
B) $\sum _ { n = 1 } ^ { 6 } \frac { 1 } { n ( n + 1 ) }$
C) $\sum _ { n = 1 } ^ { 6 } \frac { 1 } { ( n + 2 ) ! }$
D) $\sum _ { n = 1 } ^ { 4 } \frac { 1 } { ( n + 1 ) ( n + 2 ) }$
E) $\sum _ { n = 0 } ^ { 5 } \frac { 1 } { ( n + 1 ) ( n + 2 ) }$

Accepted Answer

The correct answer is A because the sum can be written as $\sum _ { n = 1 } ^ { 6 } \frac { 1 } { ( n + 1 ) ( n + 2 ) }$ using sigma notation.

Question 4

Use a calculator to find the sum.Round to four decimal places. $$\sum _ { k = 3 } ^ { 9 } \frac { 3 } { k + 1 }$$
A)5.7869
B)8.7869
C)0.3
D)3.2869
E)2.9869

Accepted Answer

Since we do not have any specific values or numbers given in the question, it is impossible to determine the answer through manual calculation. Therefore, we need to use an online calculator to find the sum of the given expression. Upon doing so, we get the answer as 3.2869, which matches with option D. Hence, D is the correct answer.

Question 5

Find the sum. $$\sum _ { k = 1 } ^ { 3 } \frac { 1 } { k ^ { 2 } + 4 }$$
A) $\frac { 209 } { 520 }$
B) $\frac { 1 } { 13 }$
C)1
D) $\frac { 107 } { 210 }$
E) $\frac { 12 } { 11 }$

Accepted Answer

The sum can be calculated by plugging in the values of $k$ from 1 to 3 into the given formula and adding the results:$$\sum _ { k = 1 } ^ { 3 } \frac { 1 } { k ^ { 2 } + 4 } = \frac { 1 } { 1 ^ { 2 } + 4 } + \frac { 1 } { 2 ^ { 2 } + 4 } + \frac { 1 } { 3 ^ { 2 } + 4 } = \frac { 1 } { 5 } + \frac { 1 } { 8 } + \frac { 1 } { 13 }.$$To find the sum, we need to find a common denominator and add the fractions:$$\frac { 1 } { 5 } + \frac { 1 } { 8 } + \frac { 1 } { 13 } = \frac { 8 \cdot 13 + 5 \cdot 13 + 5 \cdot 8 } { 5 \cdot 8 \cdot 13 } = \frac { 104 + 65 + 40 } { 520 } = \frac { 209 } { 520 }.$$

Question 6

Find the indicated partial sum of the series. $$\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 3 } \right) ^ { i }$$ fourth partial sum
A) $$\frac { 2 } { 81 }$$
B) $$\frac { 242 } { 81 }$$
C) $\frac { 80 } { 81 }$ ​
D) $\frac { 26 } { 81 }$ ​
E) $\frac { 82 } { 81 }$

Accepted Answer

The series is a geometric series with the first term $a = 2\left(\frac{1}{3}\right)^1 = \frac{2}{3}$ and common ratio $r = \frac{1}{3}$. The formula for the sum of the first $n$ terms of a geometric series is $S_n = a\frac{1-r^n}{1-r}$. For the fourth partial sum ($n=4$), we have $S_4 = \frac{2}{3}\frac{1-(\frac{1}{3})^4}{1-\frac{1}{3}} = \frac{2}{3}\frac{1-\frac{1}{81}}{\frac{2}{3}} = \frac{2}{3} \cdot \frac{80}{81} \cdot \frac{3}{2} = \frac{80}{81}$.

Question 7

Evaluate the sum.​ $$\sum _ { k = 1 } ^ { 6 } 5 k$$ ​
A)105
B)97
C)30
D)28
E)210

Accepted Answer

The sum can be calculated directly: $5(1) + 5(2) + 5(3) + 5(4) + 5(5) + 5(6) = 5 + 10 + 15 + 20 + 25 + 30 = 105.$

Question 8

Evaluate the sum.​ $$\sum _ { k = 2 } ^ { 5 } 2 k$$ ​
A)28
B)5
C)2
D)56
E)none of these

Accepted Answer

The sum can be calculated as follows: $2(2) + 2(3) + 2(4) + 2(5) = 4 + 6 + 8 + 10 = 28.$

Quiz 52: Sequences and Series

Assume that the sequence is defined recursively.Find the first three terms of the sequence. $a _ { 1 } = 3 \text { and } a _ { n + 1 } = 4 a _ { n } ^ { 2 }$

Use sigma notation to write the sum. $\frac { 1 } { 3 \cdot 2 } + \frac { 1 } { 4 \cdot 3 } + \ldots + \frac { 1 } { 8 \cdot 7 }$

Use a calculator to find the sum.Round to four decimal places. $\sum _ { k = 3 } ^ { 9 } \frac { 3 } { k + 1 }$

Find the sum. $\sum _ { k = 1 } ^ { 3 } \frac { 1 } { k ^ { 2 } + 4 }$

Find the indicated partial sum of the series. $\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 3 } \right) ^ { i }$ fourth partial sum

Evaluate the sum. $\sum _ { k = 1 } ^ { 6 } 5 k$

Evaluate the sum. $\sum _ { k = 2 } ^ { 5 } 2 k$

Quiz 52: Sequences and Series

Assume that the sequence is defined recursively.Find the first three terms of the sequence.​ a1=3 and an+1=4an2a _ { 1 } = 3 \text { and } a _ { n + 1 } = 4 a _ { n } ^ { 2 }a1​=3 and an+1​=4an2​ ​

Use sigma notation to write the sum. 13⋅2+14⋅3+…+18⋅7\frac { 1 } { 3 \cdot 2 } + \frac { 1 } { 4 \cdot 3 } + \ldots + \frac { 1 } { 8 \cdot 7 }3⋅21​+4⋅31​+…+8⋅71​

Use a calculator to find the sum.Round to four decimal places. ∑k=393k+1\sum _ { k = 3 } ^ { 9 } \frac { 3 } { k + 1 }∑k=39​k+13​

Find the sum. ∑k=131k2+4\sum _ { k = 1 } ^ { 3 } \frac { 1 } { k ^ { 2 } + 4 }∑k=13​k2+41​

Find the indicated partial sum of the series. ∑i=1∞2(13) i\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 3 } \right) ^ { i }∑i=1∞​2(31​) i fourth partial sum

Evaluate the sum.​ ∑k=165k\sum _ { k = 1 } ^ { 6 } 5 k∑k=16​5k ​

Evaluate the sum.​ ∑k=252k\sum _ { k = 2 } ^ { 5 } 2 k∑k=25​2k ​