Numerical sequences: math exercises in 1st grade corrected in PDF.

A series of math exercises for 1st grade on numerical sequences.

You will find in these cards on the numerical series in the first year of secondary school, the following concepts:

  1. definition of a numerical sequence;
  2. arithmetic sequence;
  3. term of rank n of an arithmetic sequence and sum of the first terms of a numerical sequence;
  4. term of rank n of a geometric sequence and sum of the first terms of a geometric sequence;
  5. direction of variation of a numerical sequence (increasing and decreasing or monotonic sequence).

A numerical sequence is a sequence of elements (usually numbers) that are arranged according to a certain construction rule.

Exercise 1:

Let (U_n) be the sequence defined for all n\in\,\mathbb{N} by u_n=2n+3.

Calculate u_0,u_1 and u_2.

Exercise 2:

Let (U_n) be the sequence defined for all n\in\,\mathbb{N} by u_n=\frac{n+1}{2n-3}.

Calculate u_0 and u_{10}.

Exercise 3:

Consider the sequence (U_n) defined for all n\in\,\mathbb{N} by u_n=2n-1.

Express u_{n+1},u_{n-1},u_{2n} and u_n+1 as functions of n.

Exercise 4:

Consider the sequence (U_n) defined by u_0=2 and, for all n\in\,\mathbb{N}, u_{n+1}=\frac{2u_n-2}{u_n-3}.

1) Calculate u_1 and u_2.

2)Using the calculator, give an approximate value of u_{15} to the nearest 10^{-2}.

Exercise 5:

Let (U_n) be the sequence defined for all n\in\,\mathbb{N} by u(n)=f(n).

The representative curve of the function f is given below.

Determine the value of the first five terms of the sequence (u_n).

Suites and functions

Exercise 6:

Let (U_n) be an arithmetic sequence of reason 2 and first term u_0=-3.

1)Express u_n as a function of n.

2)Calculate u_{20}.

Exercise 7:

Are the following sequences arithmetic? Justify.

a)(U_n) defined by u_0=2 and, for any n\in\mathbb{N},u_{n+1}=u_n-4.

b)(V_n) defined for all n\in\,\mathbb{N} by v_n=-n+3.

c)(W_n) defined for all n\in\,\mathbb{N} by w_n=n^2-3.

Exercise 8:

Are the following sequences geometric? Justify.

a)(U_n) defined by u_0=2 and, for any n\in\mathbb{N},u_{n+1}=\frac{u_n}{2}.

b)(V_n) defined for any n\in\,\mathbb{N} by v_n=-3^n.

c)(W_n) defined for all n\in\,\mathbb{N} by w_n=\frac{1}{4^n}.

Exercise 9:

Yacine prepared a chocolate cake and put it on the table
in a plate in the kitchen. Every time he passes
in front, he helps himself to half of what is left.

Numerical sequences

We note u_n, the proportion of the cake which remains in the plate
after Yacine has served himself n times .

1. Give the value of u_0 and u_1.

2. Justify that the sequence (U_n) is a geometric sequence and specify its reason.

Exercise 10:

By studying the sign of u_{n+1}-u_n, study the variations of the series (u_n),

defined for all n\in\,\mathbb{N}.

a)u_n=n^2+2n.

b)u_n=\frac{4}{n+1}.

c)u_n=-5^n.

Exercise 11:

Let (U_n) be the sequence defined for any integer n\geq\,,1 by u_n=\frac{2^n}{n}.

1)Calculate \frac{u_{n+1}}{u_n}.

2)Solve the inequation \frac{2n}{n+1}>1.

3)Deduce the variations of the sequence (u_n).

Exercise 12:

Yanis has a large collection of Russian dolls.

We are interested in a series of Russian dolls.

The smallest figurine is 1 cm high.

Each doll is inside a doll that is 0.5 cm taller than her.

We note u_n, the size of the n-th doll (in ascending order).
So we have u_1=1.

1. Express u_n as a function of n.

2. What is the size of the 10th doll?

3. If, instead of nesting the dolls, they were stacked, what would be the
would be the height of a stack of 10 dolls?

Russian dolls

Exercise 13:

1.let (u_n) be the sequence defined for all n\in\,\mathbb{N} by u_n=2n+3.
Calculate u_0, u_1 and u_2.

2. Let (u_n) be the sequence defined all n\in\,\mathbb{N} by u_n=\frac{n+1}{2n-3}
Calculate u_0 and u_{10}.

3. Consider the sequence (u_n) defined for all n\in\,\mathbb{N} by u_n=2^n-1.
Calculate the first five terms of the sequence (u_n).

4. Consider the sequence (u_n) defined for all n\in\,\mathbb{N} by u_n=2n-1.
Express u_{n+1};u_{n-1};u_{2n};u_{n}+1 as a function of n.

5. Let (u_n) be the sequence defined for all n\in\,\mathbb{N} by u_n=n^2+1.
Express u_{n+1};u_{n-1};u_{2n};u_{n}+1 as a function of n.

Exercise 14:

One morning, Mathéo decides to put a container in his garden, containing 200 g of hazelnuts.
Every afternoon, a squirrel comes and eats half of the container, then Mathéo puts 80 g
of hazelnuts in the evening.
We note u_n the quantity in grams of hazelnuts in the container on the n-th day in the morning.
1. Give the value of u_1 and u_2.
2. Express u_{n+l}as a function of u_n.

squirrel and numeric suites

Exercise 15:

Let (u_n) be the sequence defined for all n\in\,\mathbb{N} by u_n\,=\,f(n).
The representative curve of the function f is given below.
Determine the value of the first five terms of the sequence (u_n).

Function curves and sequences

Exercise 16:

Let (v_n) be the sequence defined by v_0\,=\,1 and, for any n\in\,\mathbb{N} by v_{n+1}\,=\,f(v_n).
The representative curve of the function f is given below.
Determine the value of the first five terms of the sequence (v_n).

Functions and sequences

Exercise 17:

Let (u_n) be an arithmetic sequence of reason 4 and first term u_0\,=\,2.
Calculate u_1,u_2 and u_3.
Let (u_n) be an arithmetic sequence of reason 2 and first term u_0\,=\,-3.
1. Express u_n as a function of n.
2.Calculate u_{20}.

Exercise 18:

By studying the sign of u_{n+1}-\,u_n, study the variations of the sequences (u_n) defined for all n\,\in\,\mathbb{N}.

a)u_n=n^2+2n\\b)u_n=\frac{4}{n+1}\\c)u_n=-5^n

Exercise 19:

Let (u_n) be the sequence defined for any integer n\geq\,\,1 by u_n=\frac{2^n}{n}.
1. Calculate \frac{u_{n+1}}{u_n}.
2. Solve the inequation \frac{2n}{n+1}>1.
3. Deduce the variations of the sequence (u_n).

Exercise 20:

Step O: Valentine draws a three-petal rosette.
Step 1: She decides to further decorate her rose window and adds a petal between two petals
consecutive.

At each step, she adds a petal between two consecutive petals.
We note u_n the number of petals the step n.
We have .
1. Draw the rosette in step 2.
2. Deduce the value of u_1 and u_2.
3. Express u_{n+1} as a function of u_n.
4. Deduce the expression of u_n.

Geometric figures

Exercise 21:

We are interested in a square sheet of paper of side 20 cm.
At each step, we fold the corners of this sheet to obtain a new square.

Square

We want to study the sequence (u_n) which corresponds to the length of the sides of the square at step n, in cm.

We have .

1. Determine the value of u_1.
2. Determine a relationship between u_n and u_{n+1}.
3. Deduce the variations of the sequence (u_n).
4. Conjecture the limit of the sequence
We now want to study the sequence (v_n) which corresponds to the thickness of the folding, in m, at step n.
The initial sheet of paper is 0.1 mm thick.
5. Determine the value of v_0 and v_1.
6. Determine a relationship between v_n and v_{n+1}.
7. Deduce the variations of the sequence (v_n).
8. Deduce the expression of v_n as a function of n.
9. Using the calculator, determine the number of steps it would take for the folding to reach the height of the Eiffel Tower, i.e. 324 m.

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