 # Numerical sequences: corrected high school math exercises in PDF.

A series of math exercises for the final year of high school on numerical sequences.

This sheet involves the following concepts:

1. definition of a sequence;
2. sum of the terms of a sequence;
3. convergence of a numerical sequence;
4. asymptotic behavior of a sequence;
5. study suite and functions;
6. recurring suites.

Exercise #1:

u is the sequence defined by and, for any natural number n, .

With the spreadsheet, the first values of and were obtained below. 1. Conjecture an expression for as a function of n.
2. Validate this conjecture by reasoning by recurrence.

Exercise #2:

V is the sequence defined by and for any natural number n, .

Prove by recurrence that for any natural number n, .

Exercise #3:

Show by recurrence that, for any natural number n, .

Exercise #4:

In this figure:

• • • the triangles are rectangles. Prove by recurrence that for any natural number n, .

Exercise #5:

Study, justifying, the limit in infinity of each of the following numerical series: Exercise #6:

u is the geometric sequence of reason 0.8 and first term .

1. For any non-zero natural number n, express as a function of n.
2. Study the limit of the sequence .

Exercise #7:

Consider the sequence defined by and for all , .

1)Let f be the function defined on by .

a) Study the variations of f on .

b) Deduce that if , then f ‘ (x) .

2)Prove by recurrence that, for any natural number n, .

3)Determine the direction of variation of the sequence .

Exercise #8:

The sequence is defined by and for all , .

1)Using a calculator or spreadsheet, determine the first ten

terms of the sequence .

2)a)What conjecture can be made about the expression of as a function of n?

b) Prove this conjecture by recurrence.

Exercise #9:

Show by recurrence that, for any natural number n not zero, .

Exercise #10:

Determine the limit of defined on using general theorems. . . . Exercise #11:

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

Let be the sequence defined for any natural number n by : .

1) Show that the sequence is geometric of reason .

Specify the first term.

2) Determine the expression of as a function of n and deduce that,

for any natural number n : .

3) Determine the limit of the sequence .

Exercise #12:

Investigate whether the following sequences, defined on , are bounded. . . .

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