Question 1

Find the sum of the infinite series. $\sum _ { i = 1 } ^ { \infty } 2 \left( - \frac { 1 } { 4 } \right) ^ { i }$
A)undefined
B) $- \frac { 2 } { 3 }$
C)2
D) $\frac { 4 } { 5 }$
E) $- \frac { 2 } { 5 }$

Accepted Answer

The series is a geometric series with first term $a = 2 \left( -\frac{1}{4} \right) = -\frac{1}{2}$ and common ratio $r = -\frac{1}{4}$. The sum of an infinite geometric series is given by $S = \frac{a}{1-r}$, provided $|r| < 1$. Here, $S = \frac{-\frac{1}{2}}{1 - (-\frac{1}{4})} = \frac{-\frac{1}{2}}{\frac{5}{4}} = -\frac{2}{5}$.

Question 2

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 $a = 2(\frac{1}{3})$ and common ratio $r = \frac{1}{3}$. The fourth partial sum of a geometric series is given by $S_n = a\frac{1-r^n}{1-r}$ for $n=4$, so $S_4 = 2(\frac{1}{3})\frac{1-(\frac{1}{3})^4}{1-\frac{1}{3}} = \frac{2}{3}\frac{1-\frac{1}{81}}{\frac{2}{3}} = \frac{80}{81}$.

Question 3

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 4

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), 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 5

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 $\sum _ { n = 1 } ^ { 6 } \frac { 1 } { ( n + 1 ) ( n + 2 ) }$.

Quiz 9: Sequences Series and Probability

Find the sum of the infinite series. ∑i=1∞2(−14) i\sum _ { i = 1 } ^ { \infty } 2 \left( - \frac { 1 } { 4 } \right) ^ { i }∑i=1∞​2(−41​) i

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

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​

Find the sum of the infinite series. $\sum _ { i = 1 } ^ { \infty } 2 \left( - \frac { 1 } { 4 } \right) ^ { i }$

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

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. $\frac { 1 } { 3 \cdot 2 } + \frac { 1 } { 4 \cdot 3 } + \ldots + \frac { 1 } { 8 \cdot 7 }$