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:

- definition of a numerical sequence;
- arithmetic sequence;
- term of rank n of an arithmetic sequence and sum of the first terms of a numerical sequence;
- term of rank n of a geometric sequence and sum of the first terms of a geometric sequence;
- 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 be the sequence defined for all by .

Calculate and .

Exercise 2:

Let be the sequence defined for all by .

Calculate and .

Exercise 3:

Consider the sequence defined for all by .

Express and as functions of n.

Exercise 4:

Consider the sequence defined by and, for all , .

1) Calculate and .

2)Using the calculator, give an approximate value of to the nearest .

Exercise 5:

Let be the sequence defined for all by .

The representative curve of the function f is given below.

Determine the value of the first five terms of the sequence .

Exercise 6:

Let be an arithmetic sequence of reason 2 and first term .

1)Express as a function of n.

2)Calculate .

Exercise 7:

Are the following sequences arithmetic? Justify.

a) defined by and, for any .

b) defined for all by .

c) defined for all by .

Exercise 8:

Are the following sequences geometric? Justify.

a) defined by and, for any .

b) defined for any by .

c) defined for all by .

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.

We note , the proportion of the cake which remains in the plate

after Yacine has served himself n times .

1. Give the value of and .

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

Exercise 10:

By studying the sign of , study the variations of the series ,

defined for all .

.

.

.

Exercise 11:

Let be the sequence defined for any integer by .

1)Calculate .

2)Solve the inequation .

3)Deduce the variations of the sequence .

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 , the size of the n-th doll (in ascending order).

So we have .

1. Express 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?

Exercise 13:

1.let be the sequence defined for all by .

Calculate , and .

2. Let be the sequence defined all by

Calculate and .

3. Consider the sequence defined for all by .

Calculate the first five terms of the sequence .

4. Consider the sequence defined for all by .

Express as a function of n.

5. Let be the sequence defined for all by .

Express 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 the quantity in grams of hazelnuts in the container on the n-th day in the morning.

1. Give the value of and .

2. Express as a function of .

Exercise 15:

Let be the sequence defined for all by .

The representative curve of the function f is given below.

Determine the value of the first five terms of the sequence .

Exercise 16:

Let be the sequence defined by and, for any by .

The representative curve of the function f is given below.

Determine the value of the first five terms of the sequence .

Exercise 17:

Let be an arithmetic sequence of reason 4 and first term .

Calculate and .

Let be an arithmetic sequence of reason 2 and first term .

1. Express as a function of n.

2.Calculate .

Exercise 18:

By studying the sign of , study the variations of the sequences defined for all

Exercise 19:

Let be the sequence defined for any integer by .

1. Calculate .

2. Solve the inequation .

3. Deduce the variations of the sequence .

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 the number of petals the step n.

We have .

1. Draw the rosette in step 2.

2. Deduce the value of and .

3. Express as a function of .

4. Deduce the expression of .

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.

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

We have .

1. Determine the value of .

2. Determine a relationship between and .

3. Deduce the variations of the sequence .

4. Conjecture the limit of the sequence

We now want to study the sequence 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 and .

6. Determine a relationship between and .

7. Deduce the variations of the sequence .

8. Deduce the expression of 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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