
The OdysseyI think Odysseus's plan to escape Polyphemus is pretty ingenious. He first considers killing the Cyclops by stabbing him in the liver while he sleeps, but then Odysseus correctly realizes that...

The Lady or the Tiger?Frank Stockton's short story "The Lady or the Tiger" is about a "semibarbaric" king who believes he has invented the perfect system of justice. The most important crimes in the kingdom are decided...

HistoryThe social and political changes of the eighteenth and early nineteenth centuries have their roots in the devastating religious wars that wracked Europe in the seventeenth century. With the rise of...

ScienceThe collision theory states that a reaction occurs when the collision between molecules cause old bonds to become broke and new bonds to form. However, the collisions must occur with enough energy,...

LiteratureThe term pothouse, or pothouse, is basically a synonym for a tavern, though some dictionaries may specify a “small” tavern. According to MerriamWebster the first known use was in the year...

Science"A 1000 W microwave" means that the microwave uses the power of 1000 Watts when it is on. Power is the rate of the consumption of energy: `P = (Delta E)/(Delta t)` , assuming the energy is consumed...

To Kill a MockingbirdIn typical Aunt Alexandra fashion, she starts barking out orders the second she walks in the door of the Finch’s house. Aunt Alexandra is asked by Atticus to help by staying with the children...

ScienceA runner runs 100 meters in 12 seconds and we are asked to find the average speed: The average speed is the total distance divided by the total time over some interval. (This is the rate of change...

ReferenceThink of it this way. Did the dog need his imagination in order to survive the frigid cold? The dog relied solely on his instincts and knowledge of nature to make it out alive. The man, however,...

PsychologyBesides the individual personal response (which can only be supplied by the asker of this question and/or the answerer), a psychologically sound answer would address the issues that were causing...

CranesThis story takes place in Korea and is set during the Korean War. The war began in 1950 and ended in 1953. At the end of World War II, in 1945, the Soviet Union took control of Korea north of the...

BusinessThe communication process is a cyclical process continually repeating until any of the steps in the process degrades to the point of destroying the communication link. To categorize the appropriate...

MathGiven 1, 1/2, 1/6, 1/24, 1/120 In this sequence the numerator of each term is 1. The terms in the denominator follow the n! (factorial) pattern. The expression for the nth them of the sequence is...

MathThe `n`th term is `a_n=(1)^(n1)` Let us check: `a_1=(1)^0=1` `a_2=(1)^1=1` `a_3=(1)^2=1` `a_4=(1)^3=1` The solution is obvious if we know that `(1)^n=1` when `n` is even and `(1)^n=1`...

MathLet us start with the sequence `1,1,1,1,1,1,...` The `n`th term of this sequence is `(1)^n` (for odd `n` we get `1` while for even `n` we get `1`) If we add `2` to each term of this sequence...

MathLet us first write the first two terms as fractions as well. `1/1,3/1,3^2/2,3^3/6,3^4/24,3^5/120,...` In the numerator we have powers of 3 (`3^0=1` and `3^1=3`). In the denominator we have...

Math`a_1=2^1/3^1=2/3` `a_2=2^2/3^2=4/9` `a_3=2^3/3^3=8/27` `a_4=2^4/3^4=16/81` `a_5=2^5/3^5=32/243`

Math`a_1=1/1^(3/2)=1/1=1` `a_2=1/2^(3/2)=1/sqrt(8)=1/(2sqrt2)` `a_3=1/3^(3/2)=1/sqrt27=1/(3sqrt3)` `a_4=1/4^(3/2)=1/sqrt64=1/(4sqrt4)` `a_5=1/5^(3/2)=1/sqrt125=1/(5sqrt5)`

MathThis is a constant sequence, meaning that all the terms are the same. Therefore, the first five terms are the same. In this case they are equal to `2/3.` `a_1=2/3` `a_2=2/3` `a_3=2/3` `a_4=2/3`...

Math`a_1=1+(1)^1=11=0` `a_2=1+(1)^2=1+1=2` `a_3=1+(1)^3=11=0` `a_4=1+(1)^4=1+1=2` `a_5=1+(1)^5=11=0` This sequence alternates between 0 and 2.

Math`a_1=1(11)(12)=1cdot0cdot(1)=0` `a_2=2(21)(22)=2cdot1cdot0=0` `a_3=3(31)(32)=3cdot2cdot1=6` `a_4=4(41)(42)=4cdot3cdot2=24` `a_5=5(51)(52)=5cdot4cdot3=60`

Math`a_1=1(1^26)=5` `a_2=2(2^26)=2(2)=4` `a_3=3(3^26)=3cdot3=9` `a_4=4(4^26)=4cdot10=40` `a_5=5(5^26)=5cdot19=95`

Math`a_n=(1)^n(n/(n+1))` `a_1=(1)^1(1/(1+1))=(1)(1/2)=1/2` Plug in n=2, to get the 2nd term `a_2=(1)^2(2/(2+1))=(1)(1)(2/3)=2/3` Plug in n=3, to get the 3rd term...

Math`a_n=(1)^(n+1)/(n^2+1)` `a_1=(1)^(1+1)/(1^2+1)=1/2` `a_2=(1)^(2+1)/(2^2+1)=1/5` `a_3=(1)^(3+1)/(3^2+1)=1/10` `a_4=(1)^(4+1)/(4^2+1)=1/17` `a_5=(1)^(5+1)/(5^2+1)=1/26` ` ` ` `

MathIt is obvious that each term is greater by four than the previous term which implies that this is arithmetic sequence. And since the first term is 3 we have `a_n=3+4(n1)=3+4n4=1+4n` The `n`th...

MathLet us compare this sequence with the following one `1,4,9,16,25,...` This is the sequence of square numbers `n^2` and each term of our sequence is one smaller then the terms in the above sequence....

MathNumerator starts with 2 and is increased by one in each term that follows `(n+1).`Denominator starts with 3 and is increased by one in each term that follows `(n+2).` Signs alternate between + and...

MathThe expression for the apparent term is `a_n=(1)^(n1)/(2^n)` ` ` `a_1=(1)^(11)/2^1=1/2` `a_2=(1)^(21)/2^2=1/4` `a_3=(1)^(31)/2^3=1/8` `a_4=(1)^(41)/2^4=1/16`

MathThe numerators are an arithmetic sequence with the difference 1, the denominators  of the difference 2. The resulting formula is a_n= (n+1)/(2n1).

MathNumerator contains powers of 2 starting with `2^0=1` while the denominator contains powers of 3 starting with `3^1=3.` Therefore, the `n`th term is `a_n=2^(n1)/3^n`...

MathNumerator is always 1 while the denominator is a square number, thus we can write the sequence as follows `1/1^2,1/2^2,1/3^2,1/4^2,1/5^2,...` From this we see that the `n`th terms is `a_n=1/n^2`

Math`a_n=4n7` `a_1=4*17=3` `a_2=4*27=87=1` `a_3=4*37=127=5` `a_4=4*47=167=9` `a_5=4*57=207=13` So, the first five terms of the sequence are `3,1,5,9,13`

Math`3^n` is 3, 9, 27, 81 and 243. Therefore the first five terms are 21/3 = 1 and 2/3, 21/9 = 1 and 8/9, 21/27 = 1 and 26/27, 21/81 = 1 and 80/81, 21/243 = 1 and 242/243.

Math`a_n=(2)^n` `a_1=(2)^1=2` `a_2=(2)^2=2*2=4` `a_3=(2)^3=2*2*2=8` `a_4=(2)^4=2*2*2*2=16` `a_5=(2)^5=2*2*2*2*2=32` So, the first five terms of the sequence are `2,4,8,16,32`

Math`a_n=(1/2)^n` `a_1=(1/2)^1=1/2` `a_2=(1/2)^2=1/4` `a_3=(1/2)^3=1/8` `a_4=(1/2)^4=1/16` `a_5=(1/2)^5=1/32` So, the first five terms of the sequence are `1/2,1/4,1/8,1/16,1/32`

Math`a_n=n/(n+2)` `a_1=1/(1+2)=1/3` `a_2=2/(2+2)=2/4=1/2` `a_3=3/(3+2)=3/5` `a_4=4/(4+2)=4/6=2/3` `a_5=5/(5+2)=5/7` So, the first five terms of the sequence are `1/3,1/2,3/5,2/3,5/7`

MathThese terms are fractions, their numerators and denominators are easy to compute. The first five terms are 3, 12/11, 9/13, 24/47 and 15/37.

Math`(1)^n` is 1 for even n and 1 for odd n. So `1+(1)^n` is 2 or 0, respectively. So the first five terms are 0, 1, 0, 1/2 and 0.

MathThe first five terms are 1, 1/4, 1/9, 1/16 and 1/25.

Math`2x+6y=16` `2x3y=7` Write the system of equations as an augmented matrix. `[[2,616],[2,37]]` Multiply the second line by 1. `[[2,616],[2,37]]` Eliminate the first column....

MathrsWe began by stating our equations as follows: 3x  2y = 27 x + 3y = 13 (NB: Note that I wrote the values of the x variable below each other, values with the y variable below each other and the...

Math`x+y=4` `2x4y=34` Write the system of equations as an augmented matrix. `[[1,14],[2,434]]` Multiply the first line by a 2. `[[2,28],[2,434]] ` Eliminate the first column....

Math`x+2y3z=28` `4y+2z=0` `x+yz=5` The equations in the matrix form can be written as, `[[1,2,3,28],[0,4,2,0],[1,1,1,5]]` Add Row 1 and Row 3 `[[1,2,3,28],[0,4,2,0],[0,3,4,33]]` Multiply...

Math`3x2y+z=15` `x+y+2z=10` `xy4z=14` `A=[[3,2,1],[1,1,2],[1,1,4]]` `b=[[15],[10],[14]]` `[Ab]=[[3,2,1,15],[1,1,2,10],[1,1,4,14]]` Multiply 2nd Row by 3 and add Row 1...

MathGiven , 2x + 6y = 22, x + 2y = 9 so A = 2 6 1 2 and B = 22 9 so the augmented matrix is A^~=[A B]= 2 6 22 1 2 9 step 1....

MathGiven system of equations are 5x  5y = 5, 2x  3y = 7 so ,we get the matrices as A= 5 5 2 3 and B= 5 7 the augmented matrix [AB] = 5 5 5...

MathGiven system of equations are 8x  4y = 7, 5x + 2y = 1 so the matrix A ,B are given as foll A = 8 4 5 2 B= 7 1 so the augmented matrix is [AB] = 8 4 7...

MathGiven system of equations are , x  3y = 5, 2x + 6y = 10 so the matrix A,B are as follows, A = 1 3 2 6 and B = 5 10 so the augmented matrix is [AB] = 1 3 5...

MathThe order of a matrix is determined by the number of rows and the number of columns. This matrix has 2 rows and 2 columns. Therefore the order of the matrix is 2 X 2.

MathThe order of a matrix is determined by the number of rows and the number of columns. This matrix has 2 rows and 3 columns. Therefore the order of the matrix is 2 X 3.
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