Online Exercises with Solutions on Numerical Sequences

The online exercises on numerical sequences presented here aim to strengthen the skills of high school students within the U.S. mathematics curriculum. The variety of exercises covers a wide range of concepts and methods, contributing to a thorough understanding of numerical sequences and their practical applications.

Exercises No. 1614 to No. 1617 focus on the calculation of terms of sequences defined by specific functions. For example, No. 1614 uses a rational fractional function to define the sequence, while No. 1615 and No. 1616 respectively use linear functions and power functions. These exercises allow high school students to practice directly calculating sequence terms, enhancing their algebraic skills and understanding of functions.

Exercises No. 1618 and No. 1619 introduce sequences defined by recurrence. Exercise No. 1618 uses a linear recurrence function to define the sequence, while No. 1619 uses a quadratic function. These exercises help students understand recurrence mechanisms and develop methods to solve sequences defined by recurrence.

Exercises No. 1620 and No. 1621 address the algebraic expression of sequence terms. Students are asked to express the terms of the sequence as a function of n, enhancing their ability to manipulate algebraic expressions and understand the relationships between sequence terms.

Exercises No. 1622 and No. 1623 explore the direction of variation of numerical sequences. Exercise No. 1622 asks whether a sequence is increasing or decreasing, while No. 1623 integrates the use of fractions. These exercises are essential for understanding the behavior and evolution of sequences.

Exercises No. 1624 and No. 1625 focus on arithmetic and geometric sequences, asking students to determine the nature of the sequence and calculate the ratios. These exercises help students consolidate their understanding of the fundamental concepts of arithmetic and geometric sequences.

Exercises No. 1626 to No. 1628 deal with the calculation of terms and sums of arithmetic and geometric sequences. Exercise No. 1626 asks students to calculate the terms of an arithmetic sequence based on its ratio and initial term, while No. 1627 does the same for a geometric sequence. Exercise No. 1628 introduces the calculation of the sum of the terms of an arithmetic sequence, a key concept for high school students.

Finally, Exercises No. 1629 and No. 1630 focus on the calculation of sums of arithmetic and geometric sequences. These exercises allow high school students to practice summing sequences, consolidating their understanding of the concepts and techniques needed to solve complex problems.

These exercises cover a wide range of mathematical concepts relevant to the high school curriculum in the United States, offering students a valuable opportunity to practice and master numerical sequences in a varied and practical context.

17 exercises

Exercise example N°1614 :

    Let the sequence (`u_(n)`) be defined for any natural number n by `u_(n)=(-5-4*n)/(4+3*n)`.
  1. Compute `u_(0)`
  2. Compute `u_(1)`

numerical sequences 11th Grade sequence

The purpose of this exercise on numerical sequences is to calculate terms of a sequence defined from a rational fraction function.

Exercise example N°1615 :

    Let the sequence (`u_(n)`) be defined for any natural number n by `u_(n)=-4-4*n`.
  1. Compute `u_(3)`
  2. Compute `u_(7)`

numerical sequences 11th Grade sequence

The purpose of this exercise on numerical sequences is to calculate terms of a sequence defined by a linear function.

Exercise example N°1616 :

    Let the sequence (`u_(n)`) be defined for any natural number n by `u_(n)=(-1)^n*4^(n+1)`.
  1. Compute `u_(1)`
  2. Compute `u_(2)`

numerical sequences 11th Grade sequence

The purpose of this exercise on numerical sequences is to calculate terms of a sequence defined by a power function.

Exercise example N°1617 :

    Let the sequence (`u_(n)`) be defined for any natural number n by `u_(n)=sqrt(3+3*n)/(5+3*n)`.
  1. Compute `u_(4)`
  2. Compute `u_(6)`

numerical sequences 11th Grade sequence

The purpose of this exercise on numerical sequences is to calculate terms of a sequence defined from a fraction and a square root.

Exercise example N°1618 :

    Let the sequence (`u_(n)`) be defined for any natural number n by `u_(0)= 2 ` and `u_(n+1)` = `1+u_(n)`.
  1. Compute `u_(3)`
  2. Compute `u_(5)`

numerical sequences 11th Grade recursive_sequence

The purpose of this exercise on numerical sequences is to calculate terms of a sequence defined by recurrence with a linear function.

Exercise example N°1619 :

    Let the sequence (`u_(n)`) be defined for any natural number n by `u_(0)= 2 ` and `u_(n+1)` = `-2+2*u_(n)^2`.
  1. Compute `u_(2)`
  2. Compute `u_(4)`

numerical sequences 11th Grade recursive_sequence

The purpose of this exercise on numerical sequences is to calculate terms of a sequence defined by recurrence with a quadratic function.

Exercise example N°1620 :

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)`.

numerical sequences 11th Grade 12th Grade

The purpose of this exercise on numerical sequences is to write in algebraic form one of the terms of the sequence.

Exercise example N°1621 :

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

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

numerical sequences 11th Grade 12th Grade

The purpose of this exercise on numerical sequences is to write in algebraic form one of the terms of the sequence.

Exercise example N°1622 :

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?

numerical sequences 11th Grade 12th Grade

Exercise on the direction of variation of a simple numerical sequence: constant sequences, increasing sequences and decreasing sequences.

Exercise example N°1623 :

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?

numerical sequences 11th Grade 12th Grade

Exercise on the direction of variation of a numerical sequence with a fraction: constant, increasing and decreasing sequences.

Exercise example N°1624 :

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.

numerical sequences 11th Grade 12th Grade

Exercise on arithmetic sequences, on geometric sequences and on common difference and on common ratio.

Exercise example N°1625 :

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.

numerical sequences 11th Grade 12th Grade

Exercise on geometric sequences, on arithmetic sequences and their reason.

Exercise example N°1626 :

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)`

numerical sequences 11th Grade 12th Grade

This exercise allows you to practice the calculation of the terms of an arithmetic sequence from its common difference and its first term.

Exercise example N°1627 :

    "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. .
  2. Compute `u_(5)`.
"

numerical sequences 11th Grade 12th Grade

This exercise allows you to practice the calculation of the terms of a geometric sequence from its common ratio and its first term.

Exercise example N°1628 :

    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.

numerical sequences 11th Grade 12th Grade

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 example N°1629 :

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

numerical sequences 11th Grade 12th Grade

This exercise allows you to practice calculating the sum of the terms of an arithmetic sequence.

Exercise example N°1630 :

    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.

numerical sequences 11th Grade 12th Grade

This exercise allows you to practice calculating the sum of the terms of a geometric sequence from its common ratio and its first term.

The numerical sequences topic is available for : 11th Grade, 12th Grade

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