
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.

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.

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

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

Math`[[4,3,,5],[1,3,,12]]`

Math`[[1,10,2,,2],[5,3,4,,0],[2,1,0,,6]]`

Math`[[3,2,3,,20],[0,25,11,,5]]`

MathThe system of linear equations represented by the augmented matrix is 1x+2y=7 2x3y=4

MathThe system of equations represented by the augment matrix is 7x5y=0 8x+3y=2

MathThe system of equations repressed by the augmented matrix is 2x+5z=12 1y2z=7 6x+3y=2

MathThe system of equations represented by the augmented matrix is 4x5y1z=12 11x+6z=25 3x+8y=29

MathGiven the matrix [7, 0] The order of the matrix is determined by the number of rows and the number of columns. This matrix has 1 row and 2 columns. Therefore the order is 1 X 2.

MathThe order of a matrix is determined by the number of rows and the number of columns. This matrix has 1 rows and 4 columns. Therefore the order of the matrix is 1 X 4.

MathThe order of a matrix is determined by the number of rows and the number of columns. This matrix has 3 rows and 1 columns. Therefore the order of the matrix is 3 X 1.

MathThe order of a matrix is determined by the number of rows and the number of columns. This matrix has 3 rows and 4 columns. Therefore the order of the matrix is 3 X 4.

Math`lim_(x>0) (1+(3x)/25))^(1/x)` To solve, let's assign values to x that are approaching zero from the left and from the right. For x values that are approaching zero from the left: `x=0.1` `y=...

MathIn a geometric sequence, the ratio between two consecutive numbers is same. ratio r = `(a2)/(a1)` To find nth term of the sequence we use an formula. `an = a1 . r^(n1)` 1. 2500, 500, 100,........

MathHello! First, we can open the parentheses using the distributive property: 8*(cos(30°) + i*sin(30°)) = 8*cos(30°) + i*(8*sin(30°)). Formally, this is the answer already, because 8cos(30°) and...

MathHello! To find a center and a radius of a circle given by its equation, it is desirable to express this equation in the form `(xa)^2+(yb)^2=r^2.` Then the center is (a, b) and the radius is r....

MathHello! It isn't clear what the problem is. I see two variants: extract some more factors (factor completely) and, to the contrast, open the parentheses (express as a sum of monomials). Let's do...

MathSturges' rule is a way of calculating the number of bins (e.g. categories or classes) of a set of data. It is assumed that the data come from a normally distributed population. Sturges' rule states...

MathWe are given that the population mean mu=12.074, with a population standard deviation sigma=.046. We are asked to find the percentage of the population with the following properties: (a) Find...

MathA pareto chart consists of a bar chart, with bars in descending order of length, and a line graph representing the cumulative total. Please see the attached graph:
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