A numerical sequence is any application of ℕ or a part of ℕ to ℝ. The calculators provided here allow you to practice calculations on numerical sequences.
Numerical sequences : calculators
Numerical sequences : games and quizzes
Numerical sequences : exercises
- Exercise 1620, algebraic form of the terms of a sequence - numerical sequences : The purpose of this exercise on numerical sequences is to write in algebraic form one of the terms of the sequence.
- Exercise 1621, algebraic form of the terms of a sequence - numerical sequences : The purpose of this exercise on numerical sequences is to write in algebraic form one of the terms of the sequence.
- Exercise 1622, increasing sequence and decreasing sequence - numerical sequences : Exercise on the direction of variation of a simple numerical sequence: constant sequences, increasing sequences and decreasing sequences.
- Exercise 1623, increasing sequence and decreasing sequence - numerical sequences : Exercise on the direction of variation of a numerical sequence with a fraction: constant, increasing and decreasing sequences.
- Exercise 1624, arithmetic and geometric sequences and reason - numerical sequences : Exercise on arithmetic sequences, on geometric sequences and on common difference and on common ratio.
- Exercise 1626, arithmetic sequences calculation of terms and common difference - numerical sequences : This exercise allows you to practice the calculation of the terms of an arithmetic sequence from its common difference and its first term.
- Exercise 1627, geometric sequences calculation of terms and common ratio - numerical sequences : This exercise allows you to practice the calculation of the terms of a geometric sequence from its common ratio and its first term.
- Exercise 1628, sum of the terms of an arithmetic sequence - numerical sequences : This exercise allows you to practice calculating the sum of the terms of an arithmetic sequence from its common difference and its first term.
- Exercise 1629, sum of the terms of an arithmetic sequence - numerical sequences : This exercise allows you to practice calculating the sum of the terms of an arithmetic sequence.
- Exercise 1630, sum of the terms of a geometric sequence - numerical sequences : This exercise allows you to practice calculating the sum of the terms of a geometric sequence from its common ratio and its first term.
Numerical sequences : Reminder
A numerical sequence or a numerical progression is any application of ℕ or a part of ℕ to ℝ
Direction of variation of a sequence: strictly increasing sequence, strictly decreasing sequence.
- To say that the sequence (`u_(n)`) is strictly increasing means that:
For any natural number n, `u_(n+1)>u_(n)`
- To say that the sequence (`u_(n)`) is strictly decreasing means that:
For any natural number n, `u_(n)>u_(n+1)`.
To show that a sequence is increasing or decreasing:
- We can calculate the difference `u_(n+1)-u_(n)`, if this difference is positive then the sequence is increasing, otherwise it is decreasing.
- We can also, if the sequence is positive and `u_n!=0`, calculate the ratio `u_(n+1)/u_(n)`, if this ratio is greater than 1 the sequence
is increasing, otherwise it is decreasing.
Arithmetic sequences, geometric sequences
Arithmetic sequences (arithmetic progression)
To say that a sequence (`u_(n)`) is arithmetic means that there is a real r such that for any natural number n, `u_(n+1)`=`u_(n)`+r.
The real r is called the common difference of the sequence (`u_(n)`).
If (`u_(n)`) is an arithmetic sequence of first term `u_(0)`, and common difference r. Then for any natural number n, `u_(n)=u_(0)+nr`
Sum of consecutive terms of an arithmetic sequence
If S=a+...+k is the sum of p consecutive terms of an arithmetic sequence then `S = p(a+k)/2`.
We deduce that `1+2+3+...+n=n(n+1)/2`
Geometric sequences (geometric progression)
To say that a sequence (`u_(n)`) is geometric means that there is a real q such that for any natural n, `u_(n+1)`=`qu_(n)`.
The real q is called the common ratio for the sequence (`u_(n)`).
If (`u_(n)`) is a geometric sequence of first term `u_(0)`, and common ratio q. Then for any natural number n, `u_(n)=u_(0)*q^n`
Sum of consecutive terms of a geometric sequence
If S=a+...+k is the sum of p consecutive terms of a geometric sequence of common ratio q (`q != 1`) then `S = (a-k*q)/(1-q)`.
We deduce that `1+q+q^2+...+q^n=(1-q^(n+1))/(1-q)`
Numerical sequences | Matrices | Vectors | Numbers | Trigonometric functions | Equations | Statistics | Complex numbers | Fractions | Finance | Real functions | Time | Geometry | Algebraic calculation