Corrected online 12th Grade, senior year, year 13 math exercises

Exercise example :

Let the sequence (`u_(n)`) defined by `u_(n)` = `(2+n)/(2+5*n)`.

Express as a function of n the terms of `u_(n+3)`.

Exercise example :

Let the sequence (`u_(n)`) defined by `u_(n)` = `-3-3*n`.

Express as a function of n the terms of `u_(n+1)`.

Exercise example :

Let the sequence (`u_(n)`) be defined for any natural number n by `u_(0)= 3 ` and `u_(n+1)` = `-3+u_(n)`.
Is this sequence increasing or decreasing?

Exercise example :

Let the sequence (`u_(n)`) be defined for any natural number n by `u_(0)= 4 ` and `u_(n+1)` = `u_(n)/5`.
Is this sequence increasing or decreasing?

Exercise example :

Let the sequence (`u_(n)`) defined for any natural number n by `u_(0)= -3 ` and `u_(n+1)` = `-7+u_(n)`.

1. Is (`u_(n)`) an arithmetic or a geometric sequence ?
2. What is the reason of (`u_(n)`)
3. Give the expression of `u_(n)` as a function of n.

Exercise example :

Let the sequence (`u_(n)`) defined for any natural number n by `u_(0)= -1 ` and `u_(n+1)` = `-9*u_(n)`.

1. Is (`u_(n)`) an arithmetic or a geometric sequence?
2. What is the reason of (`u_(n)`).
3. Give the expression of `u_(n)` as a function of n.

Exercise example :

Let (`u_(n)`) be an arithmetic sequence of common difference -6, and of first term `u_(0)= 1 `.

1. Give the expression of `u_(n)` as a function of n.
2. Compute `u_(3)`

Exercise example :

"Let (`u_(n)`) be a geometric sequence of reason 8, and of first term `u_(0)= 2 `.
1. Give the expression of `u_(n)` as a function of n
.
2. Compute `u_(5)`.

Exercise example :

Let (`u_(n)`) be an arithmetic sequence of common difference 6, and of first term `u_(0)= 1`. Let S be the sum of `u_(3)` to `u_(25)`. S=`u_(3)`+`u_(4)`+`u_(5)`+`. . .`+`u_(25)`.
1. Compute the number of terms in S.
2. Compute S.

Exercise example :

Let S be the sum defined by S = `1`.
1. Compute the number of terms in S.
2. Compute S.

Exercise example :

Let (`u_(n)`) be a geometric sequence of common ratio -2, and of first term `u_(0)= -2 `. Let S be the sum of `u_(2)` to `u_(14)`. S=`u_(2)`+`u_(3)`+`u_(4)`+`. . .`+`u_(14)`.
1. Calculate `u_(2)`
2. Calculate `u_(14)`.
3. Deduce S.

Exercise example :

Compute the roots of P(x) =`-4+8*x+3*x^2-x^3`.

Exercise example :

Write in algebraic form the complex number Z = `(-4-5*i)/(2+3*i)`

Exercise example :

Compute the real part of the complex number Z = `(2-4*i)/(1+2*i)`

Exercise example :

Calculate the imaginary part of the complex number Z = `(1-3*i)/(5+i)`

Exercise example :

Compute the conjugate of the complex number Z = `(5-2*i)/(1+i)`

Exercise example :

z = `-3+2i`
z' = `5-4i`
Compute `z*z'`.

Exercise example :

Compute the imaginary part of the complex number, Z = `-3+2*i`

Exercise example :

Compute the real part of the complex number, Z = `-5+7*i`

Exercise example :

Represent in the complex plane, the point of affix `4+5i`.

Exercise example :

Express ln(25) as a function of ln(5) .

Exercise example :

Express `ln(1/27)` as a function of ln(3)

Exercise example :

Express `-3/8*ln(1/(27))` as a function of ln(3)

Exercise example :

Express `-5/8*ln(sqrt(2))` as a function of ln(2)

Exercise example :

Compute an antiderivative of the function `f(x)=7/(9+7*x)` on `RR^+` .

Exercise example :

Compute an antiderivative of the function `f(x)=(8*x)/(1+4*x^2)` on `RR^+` .

Exercise example :

Calculate the derivative of the function `ln(x)^5`.

Exercise example :

Calculate the derivative of the function `ln(9+9*x^2)`.

Exercise example :

Simplify the following expression `e^ln(3)+e^ln(4)`.

Exercise example :

Simplify the following expression `e^ln(8)/e^ln(4)`.

Exercise example :

Simplify the following expression `e^(ln(8)*ln(4))`.

Exercise example :

Calculate the derivative of the function `e^(3+5*x^2)`.

Exercise example :

Let f be the function defined by f(x)= `3-2*x^2+x^3` ,compute an antiderivative of f, `F(x)`, with F(x)=0.